Society for Industrial and Applied Mathematics: SIAM Journal on Numerical Analysis: Table of Contents
Table of Contents for SIAM Journal on Numerical Analysis. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/sjnaam?af=R
Society for Industrial and Applied Mathematics: SIAM Journal on Numerical Analysis: Table of Contents
Society for Industrial and Applied Mathematics
enUS
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg
https://epubs.siam.org/loi/sjnaam?af=R

Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation
https://epubs.siam.org/doi/abs/10.1137/23M1619617?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 20712086, October 2024. <br/> Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in [math], where [math], convergence of order [math] is proved in [math]. Here [math] denotes the time step size and [math] the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in [math]. The stated convergence behavior is illustrated by several numerical examples.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 20712086, October 2024. <br/> Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in [math], where [math], convergence of order [math] is proved in [math]. Here [math] denotes the time step size and [math] the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in [math]. The stated convergence behavior is illustrated by several numerical examples. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation
10.1137/23M1619617
SIAM Journal on Numerical Analysis
20240903T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Lun Ji
Alexander Ostermann
Frédéric Rousset
Katharina Schratz
Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation
62
5
2071
2086
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M1619617
https://epubs.siam.org/doi/abs/10.1137/23M1619617?af=R
© 2024 Society for Industrial and Applied Mathematics

Error Analysis Based on Inverse Modified Differential Equations for Discovery of Dynamics Using Linear Multistep Methods and Deep Learning
https://epubs.siam.org/doi/abs/10.1137/22M152373X?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 20872120, October 2024. <br/> Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates. Several numerical experiments are performed to verify the theoretical analysis.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 20872120, October 2024. <br/> Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates. Several numerical experiments are performed to verify the theoretical analysis. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Error Analysis Based on Inverse Modified Differential Equations for Discovery of Dynamics Using Linear Multistep Methods and Deep Learning
10.1137/22M152373X
SIAM Journal on Numerical Analysis
20240904T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Aiqing Zhu
Sidi Wu
Yifa Tang
Error Analysis Based on Inverse Modified Differential Equations for Discovery of Dynamics Using Linear Multistep Methods and Deep Learning
62
5
2087
2120
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/22M152373X
https://epubs.siam.org/doi/abs/10.1137/22M152373X?af=R
© 2024 Society for Industrial and Applied Mathematics

TwoScale Finite Element Approximation of a Homogenized Plate Model
https://epubs.siam.org/doi/abs/10.1137/23M1596272?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 21212142, October 2024. <br/> Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proved for a multiaffine finite element discretization of the involved threedimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the twodimensional isometryconstrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 21212142, October 2024. <br/> Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proved for a multiaffine finite element discretization of the involved threedimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the twodimensional isometryconstrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
TwoScale Finite Element Approximation of a Homogenized Plate Model
10.1137/23M1596272
SIAM Journal on Numerical Analysis
20240911T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Martin Rumpf
Stefan Simon
Christoph Smoch
TwoScale Finite Element Approximation of a Homogenized Plate Model
62
5
2121
2142
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M1596272
https://epubs.siam.org/doi/abs/10.1137/23M1596272?af=R
© 2024 Society for Industrial and Applied Mathematics

Numerical Schemes for Coupled Systems of Nonconservative Hyperbolic Equations
https://epubs.siam.org/doi/abs/10.1137/23M1615176?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 21432171, October 2024. <br/> Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Laxcurves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative product of the original system. Using an asymptotic analysis the relaxation limit and its stability are investigated in the uncoupled setting. It is shown that the pathconservative Lax–Friedrichs scheme arises from a discrete limit of an implicitexplicit scheme for the relaxation system. Employing the relaxation approach, a novel technique to couple two nonconservative systems under a large class of coupling conditions is established. A particular coupling strategy motivated from conservative Kirchhoff conditions is introduced and a corresponding Riemann solver provided. A fully discrete scheme for coupled nonconservative products is derived and studied in terms of path conservation. Numerical experiments applying the approach to a coupled model of vascular blood flow are presented.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 21432171, October 2024. <br/> Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Laxcurves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative product of the original system. Using an asymptotic analysis the relaxation limit and its stability are investigated in the uncoupled setting. It is shown that the pathconservative Lax–Friedrichs scheme arises from a discrete limit of an implicitexplicit scheme for the relaxation system. Employing the relaxation approach, a novel technique to couple two nonconservative systems under a large class of coupling conditions is established. A particular coupling strategy motivated from conservative Kirchhoff conditions is introduced and a corresponding Riemann solver provided. A fully discrete scheme for coupled nonconservative products is derived and studied in terms of path conservation. Numerical experiments applying the approach to a coupled model of vascular blood flow are presented. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Numerical Schemes for Coupled Systems of Nonconservative Hyperbolic Equations
10.1137/23M1615176
SIAM Journal on Numerical Analysis
20240911T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Niklas Kolbe
Michael Herty
Siegfried Müller
Numerical Schemes for Coupled Systems of Nonconservative Hyperbolic Equations
62
5
2143
2171
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M1615176
https://epubs.siam.org/doi/abs/10.1137/23M1615176?af=R
© 2024 Society for Industrial and Applied Mathematics

A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field
https://epubs.siam.org/doi/abs/10.1137/23M156968X?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 21722195, October 2024. <br/> Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two and threedimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimalorder convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 21722195, October 2024. <br/> Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two and threedimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimalorder convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field
10.1137/23M156968X
SIAM Journal on Numerical Analysis
20240917T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Genming Bai
Jiashun Hu
Buyang Li
A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field
62
5
2172
2195
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M156968X
https://epubs.siam.org/doi/abs/10.1137/23M156968X?af=R
© 2024 Society for Industrial and Applied Mathematics

Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes
https://epubs.siam.org/doi/abs/10.1137/24M1631614?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 21962221, October 2024. <br/> Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniformintime error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 21962221, October 2024. <br/> Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniformintime error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes
10.1137/24M1631614
SIAM Journal on Numerical Analysis
20240918T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yuanyuan Feng
Lei Li
JianGuo Liu
Tao Tang
Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes
62
5
2196
2221
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/24M1631614
https://epubs.siam.org/doi/abs/10.1137/24M1631614?af=R
© 2024 Society for Industrial and Applied Mathematics

Analysis of Local Discontinuous Galerkin Methods with ImplicitExplicit Time Marching for Linearized KdV Equations
https://epubs.siam.org/doi/abs/10.1137/24M1635818?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 22222248, October 2024. <br/> Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEXLDG schemes, combining local discontinuous Galerkin spatial discretization with implicitexplicit Runge–Kutta temporal discretization, for the linearized onedimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 22222248, October 2024. <br/> Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEXLDG schemes, combining local discontinuous Galerkin spatial discretization with implicitexplicit Runge–Kutta temporal discretization, for the linearized onedimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Analysis of Local Discontinuous Galerkin Methods with ImplicitExplicit Time Marching for Linearized KdV Equations
10.1137/24M1635818
SIAM Journal on Numerical Analysis
20240919T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Haijin Wang
Qi Tao
ChiWang Shu
Qiang Zhang
Analysis of Local Discontinuous Galerkin Methods with ImplicitExplicit Time Marching for Linearized KdV Equations
62
5
2222
2248
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/24M1635818
https://epubs.siam.org/doi/abs/10.1137/24M1635818?af=R
© 2024 Society for Industrial and Applied Mathematics

On the Optimality of TargetDataDependent Kernel Greedy Interpolation in Sobolev Reproducing Kernel Hilbert Spaces
https://epubs.siam.org/doi/abs/10.1137/23M1587956?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 22492275, October 2024. <br/> Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the [math]greedy targetdataindependent selection rule and can additionally be proven to be optimal when they fully exploit adaptivity ([math]greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in reproducing kernel Hilbert spaces, as they allow us to compare adaptive interpolation with nonadaptive best nonlinear approximation.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 22492275, October 2024. <br/> Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the [math]greedy targetdataindependent selection rule and can additionally be proven to be optimal when they fully exploit adaptivity ([math]greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in reproducing kernel Hilbert spaces, as they allow us to compare adaptive interpolation with nonadaptive best nonlinear approximation. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
On the Optimality of TargetDataDependent Kernel Greedy Interpolation in Sobolev Reproducing Kernel Hilbert Spaces
10.1137/23M1587956
SIAM Journal on Numerical Analysis
20240923T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Gabriele Santin
Tizian Wenzel
Bernard Haasdonk
On the Optimality of TargetDataDependent Kernel Greedy Interpolation in Sobolev Reproducing Kernel Hilbert Spaces
62
5
2249
2275
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M1587956
https://epubs.siam.org/doi/abs/10.1137/23M1587956?af=R
© 2024 Society for Industrial and Applied Mathematics

Numerical Reconstruction of Diffusion and Potential Coefficients from Two Observations: Decoupled Recovery and Error Estimates
https://epubs.siam.org/doi/abs/10.1137/23M1590743?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/5">Volume 62, Issue 5</a>, Page 22762307, October 2024. <br/> Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output leastsquares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in [math] for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 22762307, October 2024. <br/> Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output leastsquares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in [math] for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Numerical Reconstruction of Diffusion and Potential Coefficients from Two Observations: Decoupled Recovery and Error Estimates
10.1137/23M1590743
SIAM Journal on Numerical Analysis
20241003T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Siyu Cen
Zhi Zhou
Numerical Reconstruction of Diffusion and Potential Coefficients from Two Observations: Decoupled Recovery and Error Estimates
62
5
2276
2307
20241031T07:00:00Z
20241031T07:00:00Z
10.1137/23M1590743
https://epubs.siam.org/doi/abs/10.1137/23M1590743?af=R
© 2024 Society for Industrial and Applied Mathematics

Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions
https://epubs.siam.org/doi/abs/10.1137/23M1561920?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 14651491, August 2024. <br/> Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number bounded by [math] where [math] blows up as [math]. It is shown that the best uniform bound in [math] that one can hope for is [math]. More precisely, we show that the upper envelope of the bound [math] is [math]. What, then, can be done to obtain a preconditioner that is robust for all [math]? The solution turns out to be a relatively minor modification of the basic substructuring algorithm of [I. Babuška et al., SIAM J. Numer. Anal., 28 (1991), pp. 624–661]: one can simply augment the preconditioner with a suitable Jacobi smoothener over the coarse grid degrees of freedom. This is shown to result in a condition number bounded by [math] where the constant is independent of [math]. Numerical results are given which shows that the simple expedient of augmentation with nodal smoothening reduces the condition number by a factor of up to two orders of magnitude.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 14651491, August 2024. <br/> Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number bounded by [math] where [math] blows up as [math]. It is shown that the best uniform bound in [math] that one can hope for is [math]. More precisely, we show that the upper envelope of the bound [math] is [math]. What, then, can be done to obtain a preconditioner that is robust for all [math]? The solution turns out to be a relatively minor modification of the basic substructuring algorithm of [I. Babuška et al., SIAM J. Numer. Anal., 28 (1991), pp. 624–661]: one can simply augment the preconditioner with a suitable Jacobi smoothener over the coarse grid degrees of freedom. This is shown to result in a condition number bounded by [math] where the constant is independent of [math]. Numerical results are given which shows that the simple expedient of augmentation with nodal smoothening reduces the condition number by a factor of up to two orders of magnitude. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions
10.1137/23M1561920
SIAM Journal on Numerical Analysis
20240701T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Mark Ainsworth
Shuai Jiang
Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions
62
4
1465
1491
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1561920
https://epubs.siam.org/doi/abs/10.1137/23M1561920?af=R
© 2024 Society for Industrial and Applied Mathematics

Discrete Weak Duality of Hybrid HighOrder Methods for Convex Minimization Problems
https://epubs.siam.org/doi/abs/10.1137/23M1594534?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 14921514, August 2024. <br/> Abstract. This paper derives a discrete dual problem for a prototypical hybrid highorder method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a posteriori error estimates on regular triangulations into simplices using primaldual techniques. This motivates an adaptive meshrefining algorithm, which performs better compared to uniform mesh refinements.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 14921514, August 2024. <br/> Abstract. This paper derives a discrete dual problem for a prototypical hybrid highorder method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a posteriori error estimates on regular triangulations into simplices using primaldual techniques. This motivates an adaptive meshrefining algorithm, which performs better compared to uniform mesh refinements. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Discrete Weak Duality of Hybrid HighOrder Methods for Convex Minimization Problems
10.1137/23M1594534
SIAM Journal on Numerical Analysis
20240704T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ngoc Tien Tran
Discrete Weak Duality of Hybrid HighOrder Methods for Convex Minimization Problems
62
4
1492
1514
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1594534
https://epubs.siam.org/doi/abs/10.1137/23M1594534?af=R
© 2024 Society for Industrial and Applied Mathematics

Randomized LeastSquares with Minimal Oversampling and Interpolation in General Spaces
https://epubs.siam.org/doi/abs/10.1137/23M160178X?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 15151538, August 2024. <br/> Abstract. In approximation of functions based on point values, leastsquares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that nearoptimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math] is larger than the dimension [math] of the approximation space by a constant multiplicative ratio. On the other hand, for [math], we obtain an interpolation strategy with a stability factor of order [math]. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, TwiceRamanujan sparsifiers, in Proceedings of the FortyFirst Annual ACM Symposium on Theory of Computing, 2009, pp. 255–262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315–2336], with polynomial computational complexity.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 15151538, August 2024. <br/> Abstract. In approximation of functions based on point values, leastsquares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that nearoptimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math] is larger than the dimension [math] of the approximation space by a constant multiplicative ratio. On the other hand, for [math], we obtain an interpolation strategy with a stability factor of order [math]. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, TwiceRamanujan sparsifiers, in Proceedings of the FortyFirst Annual ACM Symposium on Theory of Computing, 2009, pp. 255–262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315–2336], with polynomial computational complexity. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Randomized LeastSquares with Minimal Oversampling and Interpolation in General Spaces
10.1137/23M160178X
SIAM Journal on Numerical Analysis
20240709T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Matthieu Dolbeault
Moulay Abdellah Chkifa
Randomized LeastSquares with Minimal Oversampling and Interpolation in General Spaces
62
4
1515
1538
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M160178X
https://epubs.siam.org/doi/abs/10.1137/23M160178X?af=R
© 2024 Society for Industrial and Applied Mathematics

Robust Finite Elements for Linearized Magnetohydrodynamics
https://epubs.siam.org/doi/abs/10.1137/23M1582783?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 15391564, August 2024. <br/> Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasirobust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for the fluid part, combined with an [math]conforming choice for the magnetic fluxes. The method introduces also a specific CIPtype stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 15391564, August 2024. <br/> Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasirobust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for the fluid part, combined with an [math]conforming choice for the magnetic fluxes. The method introduces also a specific CIPtype stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Robust Finite Elements for Linearized Magnetohydrodynamics
10.1137/23M1582783
SIAM Journal on Numerical Analysis
20240709T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
L. Beirão da Veiga
F. Dassi
G. Vacca
Robust Finite Elements for Linearized Magnetohydrodynamics
62
4
1539
1564
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1582783
https://epubs.siam.org/doi/abs/10.1137/23M1582783?af=R
© 2024 Society for Industrial and Applied Mathematics

FullSpectrum Dispersion Relation Preserving SummationbyParts Operators
https://epubs.siam.org/doi/abs/10.1137/23M1586471?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 15651588, August 2024. <br/> Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with highfrequency components. In this paper, we define and give explicit examples of interior [math]dispersionrelationpreserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dualpair finitedifference schemes for systems of hyperbolic partial differential equations which satisfy the summationbyparts principle and preserve the dispersion relation of the continuous problem uniformly to an [math] error tolerance for their interior stencil. We give a general framework to design provably stable finitedifference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ([math]mode) present on any equidistant grid at a tolerance of [math] maximum error within the interior stencil, with minimal extra stencil points. As standard finitedifference schemes have a [math] dispersion error for highfrequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 15651588, August 2024. <br/> Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with highfrequency components. In this paper, we define and give explicit examples of interior [math]dispersionrelationpreserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dualpair finitedifference schemes for systems of hyperbolic partial differential equations which satisfy the summationbyparts principle and preserve the dispersion relation of the continuous problem uniformly to an [math] error tolerance for their interior stencil. We give a general framework to design provably stable finitedifference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ([math]mode) present on any equidistant grid at a tolerance of [math] maximum error within the interior stencil, with minimal extra stencil points. As standard finitedifference schemes have a [math] dispersion error for highfrequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
FullSpectrum Dispersion Relation Preserving SummationbyParts Operators
10.1137/23M1586471
SIAM Journal on Numerical Analysis
20240711T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Christopher Williams
Kenneth Duru
FullSpectrum Dispersion Relation Preserving SummationbyParts Operators
62
4
1565
1588
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1586471
https://epubs.siam.org/doi/abs/10.1137/23M1586471?af=R
© 2024 Society for Industrial and Applied Mathematics

Localized Implicit Time Stepping for the Wave Equation
https://epubs.siam.org/doi/abs/10.1137/23M1582618?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 15891608, August 2024. <br/> Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a restart is introduced after a certain number of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 15891608, August 2024. <br/> Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a restart is introduced after a certain number of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Localized Implicit Time Stepping for the Wave Equation
10.1137/23M1582618
SIAM Journal on Numerical Analysis
20240715T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Dietmar Gallistl
Roland Maier
Localized Implicit Time Stepping for the Wave Equation
62
4
1589
1608
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1582618
https://epubs.siam.org/doi/abs/10.1137/23M1582618?af=R
© 2024 Society for Industrial and Applied Mathematics

On a New Class of BDF and IMEX Schemes for Parabolic Type Equations
https://epubs.siam.org/doi/abs/10.1137/23M1612986?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 16091637, August 2024. <br/> Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higherorder schemes, making it impractical to use highorder schemes for stiff problems. We construct in this paper a new class of BDF and implicitexplicit schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which is allowed with a usual higherorder scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second to fourthorder schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 16091637, August 2024. <br/> Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higherorder schemes, making it impractical to use highorder schemes for stiff problems. We construct in this paper a new class of BDF and implicitexplicit schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which is allowed with a usual higherorder scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second to fourthorder schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
On a New Class of BDF and IMEX Schemes for Parabolic Type Equations
10.1137/23M1612986
SIAM Journal on Numerical Analysis
20240716T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Fukeng Huang
Jie Shen
On a New Class of BDF and IMEX Schemes for Parabolic Type Equations
62
4
1609
1637
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1612986
https://epubs.siam.org/doi/abs/10.1137/23M1612986?af=R
© 2024 Society for Industrial and Applied Mathematics

Discrete Maximal Regularity for the Discontinuous Galerkin TimeStepping Method without Logarithmic Factor
https://epubs.siam.org/doi/abs/10.1137/23M1580802?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 16381659, August 2024. <br/> Abstract. Maximal regularity is a kind of a priori estimate for parabolictype equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) timestepping method. We will establish such an estimate without logarithmic factor over a quasiuniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 16381659, August 2024. <br/> Abstract. Maximal regularity is a kind of a priori estimate for parabolictype equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) timestepping method. We will establish such an estimate without logarithmic factor over a quasiuniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Discrete Maximal Regularity for the Discontinuous Galerkin TimeStepping Method without Logarithmic Factor
10.1137/23M1580802
SIAM Journal on Numerical Analysis
20240722T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Takahito Kashiwabara
Tomoya Kemmochi
Discrete Maximal Regularity for the Discontinuous Galerkin TimeStepping Method without Logarithmic Factor
62
4
1638
1659
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1580802
https://epubs.siam.org/doi/abs/10.1137/23M1580802?af=R
© 2024 Society for Industrial and Applied Mathematics

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions
https://epubs.siam.org/doi/abs/10.1137/23M1607398?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 16601686, August 2024. <br/> Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with sheardependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 16601686, August 2024. <br/> Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with sheardependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions
10.1137/23M1607398
SIAM Journal on Numerical Analysis
20240723T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Julius Jeßberger
Alex Kaltenbach
Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions
62
4
1660
1686
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1607398
https://epubs.siam.org/doi/abs/10.1137/23M1607398?af=R
© 2024 Society for Industrial and Applied Mathematics

DualityBased Error Control for the Signorini Problem
https://epubs.siam.org/doi/abs/10.1137/22M1534791?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 16871712, August 2024. <br/> Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign and boundpreserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in [math] for any [math]. We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 16871712, August 2024. <br/> Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign and boundpreserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in [math] for any [math]. We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
DualityBased Error Control for the Signorini Problem
10.1137/22M1534791
SIAM Journal on Numerical Analysis
20240723T07:00:00Z
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license
Ben S. Ashby
Tristan Pryer
DualityBased Error Control for the Signorini Problem
62
4
1687
1712
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/22M1534791
https://epubs.siam.org/doi/abs/10.1137/22M1534791?af=R
© 2024 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license

Accurately Recover Global Quasiperiodic Systems by Finite Points
https://epubs.siam.org/doi/abs/10.1137/23M1620247?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 17131735, August 2024. <br/> Abstract. Quasiperiodic systems, related to irrational numbers, are spacefilling structures without decay or translation invariance. How to accurately recover these systems, especially for lowregularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and lowregularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lowerdimensional definition domain of quasiperiodic function and the higherdimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 17131735, August 2024. <br/> Abstract. Quasiperiodic systems, related to irrational numbers, are spacefilling structures without decay or translation invariance. How to accurately recover these systems, especially for lowregularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and lowregularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lowerdimensional definition domain of quasiperiodic function and the higherdimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Accurately Recover Global Quasiperiodic Systems by Finite Points
10.1137/23M1620247
SIAM Journal on Numerical Analysis
20240724T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Kai Jiang
Qi Zhou
Pingwen Zhang
Accurately Recover Global Quasiperiodic Systems by Finite Points
62
4
1713
1735
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1620247
https://epubs.siam.org/doi/abs/10.1137/23M1620247?af=R
© 2024 Society for Industrial and Applied Mathematics

Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework
https://epubs.siam.org/doi/abs/10.1137/23M1581133?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 17361758, August 2024. <br/> Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finitedimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the onetoone correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finitedimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 17361758, August 2024. <br/> Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finitedimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the onetoone correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finitedimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework
10.1137/23M1581133
SIAM Journal on Numerical Analysis
20240725T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Francesca Scarabel
Rossana Vermiglio
Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework
62
4
1736
1758
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1581133
https://epubs.siam.org/doi/abs/10.1137/23M1581133?af=R
© 2024 Society for Industrial and Applied Mathematics

Polynomial Interpolation of Function Averages on Interval Segments
https://epubs.siam.org/doi/abs/10.1137/23M1598271?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 17591781, August 2024. <br/> Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrangetype basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 17591781, August 2024. <br/> Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrangetype basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Polynomial Interpolation of Function Averages on Interval Segments
10.1137/23M1598271
SIAM Journal on Numerical Analysis
20240725T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ludovico Bruni Bruno
Wolfgang Erb
Polynomial Interpolation of Function Averages on Interval Segments
62
4
1759
1781
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1598271
https://epubs.siam.org/doi/abs/10.1137/23M1598271?af=R
© 2024 Society for Industrial and Applied Mathematics

Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for ThinStructure Interactions
https://epubs.siam.org/doi/abs/10.1137/23M1578401?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 17821813, August 2024. <br/> Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluidstructure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thinstructure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimalorder convergence of the proposed method in the [math] norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 17821813, August 2024. <br/> Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluidstructure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thinstructure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimalorder convergence of the proposed method in the [math] norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for ThinStructure Interactions
10.1137/23M1578401
SIAM Journal on Numerical Analysis
20240730T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Buyang Li
Weiwei Sun
Yupei Xie
Wenshan Yu
Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for ThinStructure Interactions
62
4
1782
1813
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1578401
https://epubs.siam.org/doi/abs/10.1137/23M1578401?af=R
© 2024 Society for Industrial and Applied Mathematics

Discontinuous Galerkin Methods for 3D–1D Systems
https://epubs.siam.org/doi/abs/10.1137/23M1627390?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 18141843, August 2024. <br/> Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the timedependent problem, a backward Euler dG formulation is also presented and analyzed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 18141843, August 2024. <br/> Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the timedependent problem, a backward Euler dG formulation is also presented and analyzed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Discontinuous Galerkin Methods for 3D–1D Systems
10.1137/23M1627390
SIAM Journal on Numerical Analysis
20240802T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Rami Masri
Miroslav Kuchta
Beatrice Riviere
Discontinuous Galerkin Methods for 3D–1D Systems
62
4
1814
1843
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1627390
https://epubs.siam.org/doi/abs/10.1137/23M1627390?af=R
© 2024 Society for Industrial and Applied Mathematics

Learning Homogenization for Elliptic Operators
https://epubs.siam.org/doi/abs/10.1137/23M1585015?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 18441873, August 2024. <br/> Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the smallscale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, datadriven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in datadriven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of datadriven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 18441873, August 2024. <br/> Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the smallscale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, datadriven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in datadriven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of datadriven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Learning Homogenization for Elliptic Operators
10.1137/23M1585015
SIAM Journal on Numerical Analysis
20240802T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Kaushik Bhattacharya
Nikola B. Kovachki
Aakila Rajan
Andrew M. Stuart
Margaret Trautner
Learning Homogenization for Elliptic Operators
62
4
1844
1873
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1585015
https://epubs.siam.org/doi/abs/10.1137/23M1585015?af=R
© 2024 Society for Industrial and Applied Mathematics

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications
https://epubs.siam.org/doi/abs/10.1137/23M1596296?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 18741900, August 2024. <br/> Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 18741900, August 2024. <br/> Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications
10.1137/23M1596296
SIAM Journal on Numerical Analysis
20240805T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Alexey Chernov
Tùng Lê
Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications
62
4
1874
1900
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1596296
https://epubs.siam.org/doi/abs/10.1137/23M1596296?af=R
© 2024 Society for Industrial and Applied Mathematics

An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity
https://epubs.siam.org/doi/abs/10.1137/23M1615656?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 19011928, August 2024. <br/> Abstract. We propose and analyze a novel symmetric Gautschitype exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical powertype nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]potential and [math]), we establish an optimal secondorder error bound in the [math]norm. For low regularity potential and nonlinearity ([math]potential and [math]), we obtain a firstorder [math]norm error bound accompanied with a uniform [math]norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal secondorder [math]norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent longtime behavior with nearconservation of mass and energy.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 19011928, August 2024. <br/> Abstract. We propose and analyze a novel symmetric Gautschitype exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical powertype nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]potential and [math]), we establish an optimal secondorder error bound in the [math]norm. For low regularity potential and nonlinearity ([math]potential and [math]), we obtain a firstorder [math]norm error bound accompanied with a uniform [math]norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal secondorder [math]norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent longtime behavior with nearconservation of mass and energy. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity
10.1137/23M1615656
SIAM Journal on Numerical Analysis
20240806T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Weizhu Bao
Chushan Wang
An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity
62
4
1901
1928
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1615656
https://epubs.siam.org/doi/abs/10.1137/23M1615656?af=R
© 2024 Society for Industrial and Applied Mathematics

A Priori Error Estimates of a Poisson Equation with Ventcel Boundary Conditions on Curved Meshes
https://epubs.siam.org/doi/abs/10.1137/23M1582497?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 19291955, August 2024. <br/> Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a secondorder term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of highorder curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order [math] and to the finite element degree [math], whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter [math]. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters [math] and [math]. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 19291955, August 2024. <br/> Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a secondorder term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of highorder curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order [math] and to the finite element degree [math], whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter [math]. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters [math] and [math]. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
A Priori Error Estimates of a Poisson Equation with Ventcel Boundary Conditions on Curved Meshes
10.1137/23M1582497
SIAM Journal on Numerical Analysis
20240808T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Fabien Caubet
Joyce Ghantous
Charles Pierre
A Priori Error Estimates of a Poisson Equation with Ventcel Boundary Conditions on Curved Meshes
62
4
1929
1955
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1582497
https://epubs.siam.org/doi/abs/10.1137/23M1582497?af=R
© 2024 Society for Industrial and Applied Mathematics

Multistage Discontinuous Petrov–Galerkin TimeMarching Scheme for Nonlinear Problems
https://epubs.siam.org/doi/abs/10.1137/23M1598088?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 19561978, August 2024. <br/> Abstract. In this article, we employ the construction of the timemarching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive highorder multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two and threestage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time step to time step. The key point of our construction is that one of the stages can be postprocessed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second, third, and fourthorder accurate, respectively. We test the convergence in time of our methods on a 2D+time semilinear partial differential equation after a semidiscretization in space.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 19561978, August 2024. <br/> Abstract. In this article, we employ the construction of the timemarching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive highorder multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two and threestage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time step to time step. The key point of our construction is that one of the stages can be postprocessed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second, third, and fourthorder accurate, respectively. We test the convergence in time of our methods on a 2D+time semilinear partial differential equation after a semidiscretization in space. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Multistage Discontinuous Petrov–Galerkin TimeMarching Scheme for Nonlinear Problems
10.1137/23M1598088
SIAM Journal on Numerical Analysis
20240809T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Judit MuñozMatute
Leszek Demkowicz
Multistage Discontinuous Petrov–Galerkin TimeMarching Scheme for Nonlinear Problems
62
4
1956
1978
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1598088
https://epubs.siam.org/doi/abs/10.1137/23M1598088?af=R
© 2024 Society for Industrial and Applied Mathematics

Domain Decomposition Methods for the Monge–Ampère Equation
https://epubs.siam.org/doi/abs/10.1137/23M1576839?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 19792003, August 2024. <br/> Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge–Ampère equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for largescale problems that are computationally intractable using existing solution methods.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 19792003, August 2024. <br/> Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge–Ampère equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for largescale problems that are computationally intractable using existing solution methods. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Domain Decomposition Methods for the Monge–Ampère Equation
10.1137/23M1576839
SIAM Journal on Numerical Analysis
20240813T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yassine Boubendir
Jake Brusca
Brittany F. Hamfeldt
Tadanaga Takahashi
Domain Decomposition Methods for the Monge–Ampère Equation
62
4
1979
2003
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1576839
https://epubs.siam.org/doi/abs/10.1137/23M1576839?af=R
© 2024 Society for Industrial and Applied Mathematics

Positivity Preserving and Mass Conservative Projection Method for the Poisson–Nernst–Planck Equation
https://epubs.siam.org/doi/abs/10.1137/23M1581649?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 20042024, August 2024. <br/> Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the [math] projection, we construct a second order Crank–Nicolson type finite difference scheme, which is linear (exclude the very efficient [math] projection part), positivity preserving, and mass conserving. Rigorous error estimates in the [math] norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., [math] projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 20042024, August 2024. <br/> Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the [math] projection, we construct a second order Crank–Nicolson type finite difference scheme, which is linear (exclude the very efficient [math] projection part), positivity preserving, and mass conserving. Rigorous error estimates in the [math] norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., [math] projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Positivity Preserving and Mass Conservative Projection Method for the Poisson–Nernst–Planck Equation
10.1137/23M1581649
SIAM Journal on Numerical Analysis
20240820T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Fenghua Tong
Yongyong Cai
Positivity Preserving and Mass Conservative Projection Method for the Poisson–Nernst–Planck Equation
62
4
2004
2024
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1581649
https://epubs.siam.org/doi/abs/10.1137/23M1581649?af=R
© 2024 Society for Industrial and Applied Mathematics

Least Squares Approximations in Linear Statistical Inverse Learning Problems
https://epubs.siam.org/doi/abs/10.1137/22M1538600?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 20252047, August 2024. <br/> Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an illposed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finitedimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a normbased cutoff operation. Moreover, we prove that the obtained rates are minimax optimal.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 20252047, August 2024. <br/> Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an illposed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finitedimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a normbased cutoff operation. Moreover, we prove that the obtained rates are minimax optimal. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
Least Squares Approximations in Linear Statistical Inverse Learning Problems
10.1137/22M1538600
SIAM Journal on Numerical Analysis
20240822T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Tapio Helin
Least Squares Approximations in Linear Statistical Inverse Learning Problems
62
4
2025
2047
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/22M1538600
https://epubs.siam.org/doi/abs/10.1137/22M1538600?af=R
© 2024 Society for Industrial and Applied Mathematics

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms
https://epubs.siam.org/doi/abs/10.1137/23M1584502?af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/62/4">Volume 62, Issue 4</a>, Page 20482070, August 2024. <br/> Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forwardbackward optimality system, which can then be solved using a time domain decomposition. Due to the forwardbackward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 20482070, August 2024. <br/> Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forwardbackward optimality system, which can then be solved using a time domain decomposition. Due to the forwardbackward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg" alttext="cover image"/></p>
New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms
10.1137/23M1584502
SIAM Journal on Numerical Analysis
20240823T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Martin J. Gander
LiuDi Lu
New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms
62
4
2048
2070
20240831T07:00:00Z
20240831T07:00:00Z
10.1137/23M1584502
https://epubs.siam.org/doi/abs/10.1137/23M1584502?af=R
© 2024 Society for Industrial and Applied Mathematics