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2010

Volume 47, Issue 6, pp. 4021-4795


A New Class of High Order Finite Volume Methods for Second Order Elliptic Equations

Long Chen

SIAM J. Numer. Anal. 47, pp. 4021-4043 (23 pages) | Cited 1 time

Online Publication Date: January 08, 2010

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In the numerical simulation of many practical problems in physics and engineering, finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry. However, they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution. In this paper, a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods. Optimal convergence rate in $H^1$-norm for our new quadratic finite volume methods over two-dimensional triangular or rectangular grids is obtained.

Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension

Yingda Cheng and Chi-Wang Shu

SIAM J. Numer. Anal. 47, pp. 4044-4072 (29 pages)

Online Publication Date: January 08, 2010

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In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612–9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630–641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.

A Rational Interpolation Scheme with Superpolynomial Rate of Convergence

Qiqi Wang, Parviz Moin, and Gianluca Iaccarino

SIAM J. Numer. Anal. 47, pp. 4073-4097 (25 pages) | Cited 2 times

Online Publication Date: January 08, 2010

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The purpose of this study is to construct a high-order interpolation scheme for arbitrary scattered datasets. The resulting function approximation is an interpolation function when the dataset is exact, or a regression if measurement errors are present. We represent each datapoint with a Taylor series, and the approximation error as a combination of the derivatives of the target function. A weighted sum of the square of the coefficient of each derivative term in the approximation error is minimized to obtain the interpolation approximation. The resulting approximation function is a high-order rational function with no poles. When measurement errors are absent, the interpolation approximation converges to the target function faster than any polynomial rate of convergence.

Optimal Error Estimate for the Div Least-squares Method with Data $f\inL^2$ and Application to Nonlinear Problems

Zhiqiang Cai and JaEun Ku

SIAM J. Numer. Anal. 47, pp. 4098-4111 (14 pages) | Cited 1 time

Online Publication Date: January 15, 2010

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The div least-squares methods have been studied by many researchers for the second-order elliptic equations, elasticity, and the Stokes equations, and optimal error estimates have been obtained in the $H(\mathrm{div})\times H^1$ norm. However, there is no known convergence rate when the given data $f$ belongs only to $L^2$ space. In this paper, we will establish an optimal error estimate in the $L^2\times H^1$ norm with the given data $f\in L^2$ and, hence, fill a theoretical gap of least-squares methods. As a consequence of this estimate, we will provide a convergence analysis for the linearization process on solving Navier–Stokes equations, which uses the div least-squares method for solving the corresponding Stokes equations.

Fully Computable Error Bounds for Discontinuous Galerkin Finite Element Approximations on Meshes with an Arbitrary Number of Levels of Hanging Nodes

Mark Ainsworth and Richard Rankin

SIAM J. Numer. Anal. 47, pp. 4112-4141 (30 pages) | Cited 1 time

Online Publication Date: January 15, 2010

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We obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous Galerkin norm (DG-norm) of the error in first order symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG) finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and DG-norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and DG-norm of the error up to a constant and higher order data oscillation terms. We also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous Galerkin finite element approximation for all versions of the method.

A FETI-DP Formulation for the Stokes Problem without Primal Pressure Components

Hyea Hyun Kim, Chang-Ock Lee, and Eun-Hee Park

SIAM J. Numer. Anal. 47, pp. 4142-4162 (21 pages) | Cited 1 time

Online Publication Date: January 15, 2010

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A scalable FETI-DP (dual-primal finite element tearing and interconnecting) algorithm for the Stokes problem that employs a lumped preconditioner is developed and analyzed. A pair of inf-sup stable velocity and pressure finite element spaces is used to obtain a discrete problem. Differently from previous approaches, no primal pressure unknowns are selected and only velocity primal unknowns at subdomain corners are selected. This leads to a symmetric and positive definite coarse problem matrix in the FETI-DP operator, while a larger and indefinite coarse problem appears in the previous approaches. In addition, its condition number bound is proved to be the same as the FETI-DP algorithm with a lumped preconditioner for elliptic problems. Numerical results are included.

The Discrete Duality Finite Volume Method for Convection-diffusion Problems

Yves Coudière and Gianmarco Manzini

SIAM J. Numer. Anal. 47, pp. 4163-4192 (30 pages) | Cited 3 times

Online Publication Date: January 20, 2010

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In this paper we extend the discrete duality finite volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cell-based gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to show convergence of the approximation, and a global first-order convergence rate is derived. The theoretical results are confirmed by some numerical experiments.

A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

O. Angelini, C. Chavant, E. Chénier, and R. Eymard

SIAM J. Numer. Anal. 47, pp. 4193-4213 (21 pages)

Online Publication Date: January 20, 2010

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In order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids, we propose a method which locally reduces the curvature of the discrete solution where the loss of monotony is observed. The discrete solution is shown to fulfill a variational formulation, thanks to the use of Lagrange multipliers. We can then show its convergence to the solution of the continuous problem, and an error estimate is derived. A numerical method, based on Uzawa's algorithm, is shown to provide accurate and stable approximate solutions to various problems. Numerical results show the increase of precision due to the application of the method.

Robust BDDC Preconditioners for Reissner–Mindlin Plate Bending Problems and MITC Elements

L. Beirão da Veiga, C. Chinosi, C. Lovadina, and L. F. Pavarino

SIAM J. Numer. Anal. 47, pp. 4214-4238 (25 pages) | Cited 1 time

Online Publication Date: January 20, 2010

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A Balancing Domain Decomposition Method by Constraints (BDDC) is constructed and analyzed for the Reissner–Mindlin plate bending problem discretized with Mixed Interpolation of Tensorial Components (MITC) finite elements. This BDDC algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints. After the implicit elimination of the interior degrees of freedom in each subdomain, the resulting plate Schur complement is solved by the preconditioned conjugate gradient method. The preconditioner is based on the solution of local Reissner–Mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse Reissner–Mindlin plate problem for the primal degrees of freedom. The main results of the paper are the proof and numerical verification that the proposed BDDC plate algorithm is scalable, quasi-optimal, and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem, both the interface plate problem and the preconditioner are positive definite. The numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties.

Finite Element Approximations for Stokes–Darcy Flow with Beavers–Joseph Interface Conditions

Yanzhao Cao, Max Gunzburger, Xiaolong Hu, Fei Hua, Xiaoming Wang, and Weidong Zhao

SIAM J. Numer. Anal. 47, pp. 4239-4256 (18 pages) | Cited 1 time

Online Publication Date: January 20, 2010

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Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. The coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. On the interface between the matrix and conduit, Beavers–Joseph interface conditions, instead of the simplified Beavers–Joseph–Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results.

Multiscale Asymptotic Method for Maxwell's Equations in Composite Materials

Liqun Cao, Ya Zhang, Walter Allegretto, and Yanping Lin

SIAM J. Numer. Anal. 47, pp. 4257-4289 (33 pages)

Online Publication Date: January 27, 2010

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In this paper we discuss the multiscale analysis of Maxwell's equations in composite materials with a periodic microstructure. The new contributions in this paper are the determination of higher-order correctors and the explicit convergence rate for the approximate solutions (see Theorem 2.3). Consequently, we present the multiscale finite element method and derive the convergence result (see Theorem 4.1). The numerical results demonstrate that higher-order correctors are essential for solving Maxwell's equations in composite materials.

High Order Contractive Runge–Kutta Methods for Volterra Functional Differential Equations

Shoufu Li

SIAM J. Numer. Anal. 47, pp. 4290-4325 (36 pages)

Online Publication Date: January 27, 2010

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A contractivity theory of Runge–Kutta methods for stiff nonlinear Volterra functional differential equations (VFDEs) is established, by means of which a family of high order contractive Runge–Kutta methods for VFDEs are identified. The application of the general theory for stiff nonlinear VFDEs to the special cases of stiff nonlinear delay differential equations and delay integro-differential equations is also presented.

On the Fourier Extension of Nonperiodic Functions

Daan Huybrechs

SIAM J. Numer. Anal. 47, pp. 4326-4355 (30 pages) | Cited 2 times

Online Publication Date: January 27, 2010

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We obtain exponentially accurate Fourier series for nonperiodic functions on the interval $[-1,1]$ by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.

Convergence Analysis of Planewave Expansion Methods for 2D Schrödinger Operators with Discontinuous Periodic Potentials

Richard Norton and Robert Scheichl

SIAM J. Numer. Anal. 47, pp. 4356-4380 (25 pages)

Online Publication Date: January 29, 2010

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In this paper we consider the problem of computing the spectrum of a two-dimensional Schrödinger operator with discontinuous, periodic potential in two dimensions using Fourier (or planewave expansion) methods. Problems of this kind are currently of great interest in the design of new optical devices to determine band gaps and to compute localized modes in photonic crystal materials. Although Fourier methods may not be every applied mathematician's first choice for this problem because of the discontinuities in the potential, we will show here that, even though (as expected) the convergence is not exponential, the method has several desirable features that make it competitive with other discretization techniques, such as finite element methods, both with respect to implementation and convergence properties. In particular, we will prove that simple preconditioners for the system matrix are optimal leading to a computational complexity of $\mathcal{O}(N\log N)$ in the number of planewaves $N$ (using the fast Fourier transform). Moreover, we derive sharp error estimates that show that the method is essentially third order in the eigenvalues and of order $\frac{3}{2}$ in the eigenfunctions in the $H^1$-norm and $\frac{5}{2}$ in the $L^2$-norm. To improve the planewave expansion method in the case of discontinuous potentials, it has been proposed in the physics literature to replace the discontinuous potential with an effective potential that is smooth, despite the additional error this incurs. We will here answer the question whether this smoothing is worth it. In fact, our convergence analysis of the modified method provides an optimal choice for the smoothing parameter, but it also shows that the overall rate of convergence is no faster than before and so smoothing does not seem to be worth it. All the theoretical results are confirmed in our numerical experiments.

Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations

Hong-lin Liao, Zhi-zhong Sun, and Han-sheng Shi

SIAM J. Numer. Anal. 47, pp. 4381-4401 (21 pages)

Online Publication Date: January 29, 2010

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A fourth-order compact difference scheme is proposed for two-dimensional linear Schrödinger equations with periodic boundary conditions. By using the discrete energy method, it is proven that the difference scheme is uniquely solvable, unconditionally stable, and convergent. A maximum norm error estimate and thus an asymptotic expansion of the discrete solution are also obtained. Using the expansion of the difference solution, high-order approximations could be achieved by Richardson extrapolations. Extension to three-dimensional problems is also discussed. Numerical experiments are included to support the theoretical results, and comparisons with the Crank–Nicolson method are presented to show the effectiveness of our method.

On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency

Miodrag S. Petković

SIAM J. Numer. Anal. 47, pp. 4402-4414 (13 pages) | Cited 1 time

Online Publication Date: January 29, 2010

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A general class of $n$-point iterative methods for solving nonlinear equations is constructed by combining methods of Newton's type and an arbitrary two-point method of the fourth order of convergence. It is proved that these methods have the convergence order $2^n$, requiring only $n+1$ function evaluations per iteration. In this way it is demonstrated that the proposed class of methods supports the Kung–Traub hypothesis (1974) on the upper bound $2^n$ of the order of multipoint methods based on $n+1$ function evaluations. Consequently, this class possess as high as possible computational efficiency in the sense of the Kung–Traub hypothesis. Numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations.

Nonequispaced Hyperbolic Cross Fast Fourier Transform

Michael Döhler, Stefan Kunis, and Daniel Potts

SIAM J. Numer. Anal. 47, pp. 4415-4428 (14 pages)

Online Publication Date: January 29, 2010

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A straightforward discretization of problems in $d$ spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.

Error Estimates for a Finite Element Discretization of a Phase Field Model for Mixtures

Ch. Eck, B. Jadamba, and P. Knabner

SIAM J. Numer. Anal. 47, pp. 4429-4445 (17 pages)

Online Publication Date: February 03, 2010

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We derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures. Special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region. The main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system, a suitable regularity of the exact solutions, and a spectrum estimate for the operator of the Allen–Cahn equation. The spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model. This is done by a technique introduced in [X. Feng and A. Prohl, Math. Comp., 73 (2004), pp. 541–567] for phase transitions of pure materials with linear constitutive relations.

A Posteriori Error Analysis of Finite Element Methods for Reissner–Mindlin Plates

Jun Hu and Yunqing Huang

SIAM J. Numer. Anal. 47, pp. 4446-4472 (27 pages)

Online Publication Date: February 03, 2010

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This paper develops an a posteriori error control theory of finite element methods for Reissner–Mindlin plates, which states that one can derive a uniformly reliable and efficient a posteriori error estimate for a given scheme by only: (1) checking three conditions; (2) designing three functions and one parameter; (3) bounding the last three terms of the abstract estimator. We apply this theory to two classes of methods and achieve robust a posteriori error controls for them.

Shear Locking in a Plane Elasticity Problem and the Enhanced Assumed Strain Method

Dietrich Braess, Pingbing Ming, and Zhong-ci Shi

SIAM J. Numer. Anal. 47, pp. 4473-4491 (19 pages)

Online Publication Date: February 03, 2010

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The enhanced assumed strain (EAS) method is a popular tool for avoiding locking phenomena, e.g., a remedy for shear locking in plane elasticity. We consider bending-dominated problems on thin bodies which can be treated as beams and prove that the degree of approximation of the EAS method is at least as good as that of a beam model. The hypercircle method is combined with arguments of nonconforming methods.

Local Multilevel Methods for Adaptive Finite Element Methods for Nonsymmetric and Indefinite Elliptic Boundary Value Problems

Huangxin Chen and Xuejun Xu

SIAM J. Numer. Anal. 47, pp. 4492-4516 (25 pages)

Online Publication Date: February 03, 2010

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In this paper, we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems. Two types of local smoothers are constructed. One is based on the original nonsymmetric problems, and the other is defined in terms of the associated symmetric problems. It is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal, which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine. Numerical experiments are reported to confirm our theory.

Analysis of a Quadratic Programming Decomposition Algorithm

G. Bencteux, E. Cancés, W. W. Hager, and C. Le Bris

SIAM J. Numer. Anal. 47, pp. 4517-4539 (23 pages)

Online Publication Date: February 17, 2010

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We analyze a decomposition algorithm for minimizing a quadratic objective function, separable in $\mathbf{x}_1$ and $\mathbf{x}_2$, subject to the constraint that $\mathbf{x}_1$ and $\mathbf{x}_2$ are orthogonal vectors on the unit sphere. Our algorithm consists of a local step where we minimize the objective function in either variable separately, while enforcing the constraints, followed by a global step where we minimize over a subspace generated by solutions to the local subproblems. We establish a local convergence result when the global minimizers are nondegenerate. Our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint. The analysis is connected with a new domain decomposition algorithm for electronic structure calculations.

Spectral Analysis of Dirichlet–Neumann Operators and Optimized Schwarz Methods with Robin Transmission Conditions

Xuejun Xu and Lizhen Qin

SIAM J. Numer. Anal. 47, pp. 4540-4568 (29 pages) | Cited 1 time

Online Publication Date: February 17, 2010

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In this paper, the tight relationship between Dirichlet–Neumann (D-N) operators and optimized Schwarz methods with Robin transmission conditions is disclosed. We describe the spectral distribution of continuous D-N operators and give a rigorous spectral analysis of discrete D-N operators. By these results, we prove that the optimized Schwarz methods with Robin transmission conditions cannot converge geometrically in the case of continuous problems. Furthermore, we get the accurate convergence rate of the two-subdomain case. In addition, an estimation of convergence rate of the optimized Schwarz methods is presented in the general case. Most of our results are asymptotically sharp.

On the Strong Convergence of Gradients in Stabilized Degenerate Convex Minimization Problems

Wolfgang Boiger and Carsten Carstensen

SIAM J. Numer. Anal. 47, pp. 4569-4580 (12 pages)

Online Publication Date: February 17, 2010

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Infimizing sequences in nonconvex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Although those oscillations are physically meaningful, finite element approximations experience difficulty in their reconstruction. The relaxation of the nonconvex minimization problem by (semi) convexification leads to a macroscopic model for the effective energy. The resulting discrete macroscopic problem is degenerate in the sense that it is convex but not strictly convex. This paper studies a modified discretization by adding a stabilization term to the discrete energy. It will be proven that for a wide class of problems, this stabilization technique leads to strong $H^1$ convergence of the macroscopic variables even on unstructured triangulations. In contrast to the work [C. Carstensen, P. Plecháĉ, S. Bartels, and A. Prohl, Interfaces Free Bound., 6 (2004), pp. 253–269] on quasi-uniform triangulations, this paper allows for general unstructured shape-regular triangulations and so enables the use of adaptive algorithms for the stabilized formulations. (An erratum has been appended to the originally posted pdf.)

Block Monotone Iterative Method for Semilinear Parabolic Equations with Nonlinear Boundary Conditions

C. V. Pao and Xin Lu

SIAM J. Numer. Anal. 47, pp. 4581-4606 (26 pages)

Online Publication Date: February 17, 2010

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Two block monotone iterative schemes, called Jacobi and Gauss–Seidel monotone iterations, are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method. These iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations, including a comparison result between them. It is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Some error estimates and a convergence theorem are given, and various sufficient conditions for the construction of upper and lower solutions are obtained. Numerical results are presented for some physical model problems, including some problems with known continuous solutions and two problems with L-shaped and trapezoidal domains.

Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

Matthias Heinkenschloss and Dmitriy Leykekhman

SIAM J. Numer. Anal. 47, pp. 4607-4638 (32 pages)

Online Publication Date: February 19, 2010

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We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity

Roland Becker, Shipeng Mao, and Zhongci Shi

SIAM J. Numer. Anal. 47, pp. 4639-4659 (21 pages) | Cited 2 times

Online Publication Date: February 19, 2010

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In this paper, we prove convergence and quasi-optimal complexity of a simple adaptive nonconforming finite element method. In each step of the algorithm, the iterative solution of the discrete system is controlled by an adaptive stopping criterion, and the local refinement is based on either a simple edge residual or a volume term, depending on an adaptive marking strategy. We prove that this marking strategy guarantees a strict reduction of the error, augmented by the volume term and an additional oscillation term, and quasi-optimal complexity of the generated sequence of meshes.

On the Similarities Between the Quasi-Newton Inverse Least Squares Method and GMRes

Rob Haelterman, Joris Degroote, Dirk Van Heule, and Jan Vierendeels

SIAM J. Numer. Anal. 47, pp. 4660-4679 (20 pages)

Online Publication Date: February 19, 2010

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We show how one of the best-known Krylov subspace methods, the generalized minimal residual method (GMRes), can be interpreted as a quasi-Newton method and how the quasi-Newton inverse least squares method (QN-ILS) relates to Krylov subspace methods in general and to GMRes in particular when applied to linear systems. We also show that we can modify QN-ILS in order to make it analytically equivalent to GMRes, without the need for extra matrix-vector products.

Compactness Properties of the DG and CG Time Stepping Schemes for Parabolic Equations

Noel J. Walkington

SIAM J. Numer. Anal. 47, pp. 4680-4710 (31 pages) | Cited 1 time

Online Publication Date: February 19, 2010

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For a broad class of parabolic equations it is shown that numerical solutions computed using the discontinuous Galerkin or the continuous Galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation. Convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these results.

Quantization Based Filtering Method Using First Order Approximation

Afef Sellami

SIAM J. Numer. Anal. 47, pp. 4711-4734 (24 pages) | Cited 1 time

Online Publication Date: February 24, 2010

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The quantization based filtering method (see [G. Pagès and H. Pham, Bernoulli, 11 (2005), pp. 893–932; G. Pagès, H. Pham, and J. Printems, Optimal quantization methods and applications to numerical problems in finance, in Handbook of Computational and Numerical Methods in Finance, S. T. Rachev, ed., Birkhäuser, Boston, 2004, pp. 253–297]) is a grid based approximation method to solve nonlinear filtering problems with discrete time observations. It relies on off-line preprocessing of some signal grids in order to construct fast recursive schemes for filter approximation. We give here an improvement of this method by taking advantage of the stationary quantizer property. The key ingredient is the use of vanishing correction terms to describe schemes based on piecewise linear approximations. Convergence results are given and numerical results are presented for the particular cases of linear Gaussian model and stochastic volatility models.

Sharp Norm-Estimations for Moore–Penrose Inverses of Stable Perturbations of Hilbert $C^*$-Module Operators

Qingxiang Xu, Yimin Wei, and Yangyang Gu

SIAM J. Numer. Anal. 47, pp. 4735-4758 (24 pages)

Online Publication Date: February 24, 2010

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It is well known that norm upper bound $\Vert\bar{T}^{\dag}\Vert\leq\frac{\Vert T^{\dag}\Vert}{1-\Vert T^{\dag}\Vert\cdot\Vert\Delta(T)\Vert}$ plays a fundamental role in stable (resp., rank-preserving) perturbation analysis for Moore–Penrose inverses of Hilbert space operators (resp., matrices). In this paper, in the general setting of Hilbert $C^*$-module operators, we provide a new approach to the study of norm upper bounds of $\bar{T}^{\dag}$ and obtain a sharp estimation for $\Vert\bar{T}^{\dag}\Vert$. This obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the Schur complement of a positive semidefinite operator matrix. Some new norm upper bounds for Schur complements of positive semidefinite operator matrices are derived.

A Finite Element Method for Density Estimation with Gaussian Process Priors

Michael Griebel and Markus Hegland

SIAM J. Numer. Anal. 47, pp. 4759-4792 (34 pages)

Online Publication Date: February 24, 2010

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A variational problem characterizing the density estimator defined by the maximum a posteriori method with Gaussian process priors is derived. It is shown that this problem is well posed and can be solved with Newton's method. Numerically, the solution is approximated by a Galerkin/finite element method with piecewise multilinear functions on uniform grids. Error bounds for this method are given and numerical experiments are performed for one-, two-, and three-dimensional examples.

Correction to “Strict Diagonal Dominance and Optimal Bounds for the Skeel Condition Number”

Shuhuang Xiang

SIAM J. Numer. Anal. 47, pp. 4793-4795 (3 pages)

Online Publication Date: March 05, 2010

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This note is to present a correction to the proof of inequality (0.1) of Theorem 0.1 in the paper “Strict diagonal dominance and optimal bounds for the Skeel condition number” [SIAM J. Numer. Anal., 45 (2007), pp. 1107–1108].
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