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SIAM J. on Numerical Analysis

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2009

Volume 47, Issue 2, pp. 805-1600


Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces

Alan Demlow

SIAM J. Numer. Anal. 47, pp. 805-827 (23 pages) | Cited 2 times

Online Publication Date: February 06, 2009

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We define higher-order analogues to the piecewise linear surface finite element method studied in [G. Dziuk, “Finite elements for the Beltrami operator on arbitrary surfaces,” in Partial Differential Equations and Calculus of Variations, Springer-Verlag, Berlin, 1988, pp. 142–155] and prove error estimates in both pointwise and $L_2$-based norms. Using the Laplace–Beltrami problem on an implicitly defined surface $\Gamma$ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to $\Gamma$ which likewise are of arbitrary degree. Then we prove a priori error estimates in the $L_2$, $H^1$, and corresponding pointwise norms that demonstrate the interaction between the “PDE error” that arises from employing a finite-dimensional finite element space and the “geometric error” that results from approximating $\Gamma$. We also consider parametric finite element approximations that are defined on $\Gamma$ and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates.

Convergence Analysis of Projection Methods for the Numerical Solution of Large Lyapunov Equations

V. Simoncini and V. Druskin

SIAM J. Numer. Anal. 47, pp. 828-843 (16 pages) | Cited 4 times

Online Publication Date: February 06, 2009

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The numerical solution of large-scale continuous-time Lyapunov matrix equations is of great importance in many application areas. Assuming that the coefficient matrix is positive definite, but not necessarily symmetric, in this paper we analyze the convergence of projection-type methods for approximating the solution matrix. Under suitable hypotheses on the coefficient matrix, we provide new asymptotic estimates for the error matrix when a Galerkin method is used in a Krylov subspace. Numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed.

Can the Nonlocal Characterization of Sobolev Spaces by Bourgain et al. Be Useful for Solving Variational Problems?

Gilles Aubert and Pierre Kornprobst

SIAM J. Numer. Anal. 47, pp. 844-860 (17 pages) | Cited 2 times

Online Publication Date: February 06, 2009

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We question whether the recent characterization of Sobolev spaces by Bourgain, Brezis, and Mironescu (2001) could be useful to solve variational problems on $W^{1,p}(\Omega)$. To answer this, we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier. Then, for the approximated formulation, we prove existence, uniqueness, and convergence of the solution to the unique solution of the initial formulation. We show that these results can also be extended in the $BV$-case. Interestingly, this approximation leads to a unified implementation, for Sobolev spaces (including with high $p$-values) and for the $BV$ space. Finally, we show how this theoretical study can indeed lead to a numerically tractable implementation, and we give some image diffusion results as an illustration.

A Goal-Oriented Adaptive Finite Element Method with Convergence Rates

Mario S. Mommer and Rob Stevenson

SIAM J. Numer. Anal. 47, pp. 861-886 (26 pages) | Cited 2 times

Online Publication Date: February 06, 2009

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An adaptive finite element method is analyzed for approximating functionals of the solution of symmetric elliptic second order boundary value problems. We show that the method converges and derive a favorable upper bound for its convergence rate and computational complexity. We illustrate our theoretical findings with numerical results.

Practical Variance Reduction via Regression for Simulating Diffusions

G. N. Milstein and M. V. Tretyakov

SIAM J. Numer. Anal. 47, pp. 887-910 (24 pages)

Online Publication Date: February 06, 2009

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The well-known variance reduction methods—the method of importance sampling and the method of control variates—can be exploited if an approximation of the required solution is known. Here we employ conditional probabilistic representations of solutions together with the regression method to obtain sufficiently inexpensive (although rather rough) estimates of the solution and its derivatives by using the single auxiliary set of approximate trajectories starting from the initial position. These estimates can effectively be used for significant reduction of variance and further accurate evaluation of the required solution. The developed approach is supported by numerical experiments.

A Domain Decomposition Method for Computing Bivariate Spline Fits of Scattered Data

Ming-Jun Lai and Larry L. Schumaker

SIAM J. Numer. Anal. 47, pp. 911-928 (18 pages)

Online Publication Date: February 13, 2009

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A domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy, discrete least-squares, and penalized least-squares splines is described. The method is based on splitting the domain into smaller domains, solving the associated smaller fitting problems, and combining the coefficients to get a global fit. Explicit error bounds are established for how well our locally constructed spline fits approximate the global fits. Some numerical examples are given to illustrate the effectiveness of the method.

Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium

V. J. Ervin, E. W. Jenkins, and S. Sun

SIAM J. Numer. Anal. 47, pp. 929-952 (24 pages) | Cited 3 times

Online Publication Date: February 13, 2009

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In this article, we analyze the flow of a fluid through a coupled Stokes–Darcy domain. The fluid in each domain is non-Newtonian, modeled by the generalized nonlinear Stokes equation in the free flow region and the generalized nonlinear Darcy equation in the porous medium. A flow rate is specified along the inflow portion of the free flow boundary. We show existence and uniqueness of a variational solution to the problem. We propose and analyze an approximation algorithm and establish a priori error estimates for the approximation.

On Optimal Convergence Rate of the Rational Krylov Subspace Reduction for Electromagnetic Problems in Unbounded Domains

Leonid Knizhnerman, Vladimir Druskin, and Mikhail Zaslavsky

SIAM J. Numer. Anal. 47, pp. 953-971 (19 pages) | Cited 4 times

Online Publication Date: February 13, 2009

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We solve an electromagnetic frequency domain induction problem in $\mathbf{R}^3$ for a frequency interval using rational Krylov subspace (RKS) approximation. The RKS is constructed by spanning on the solutions for a certain a priori chosen set of frequencies. We reduce the problem of the optimal choice of these frequencies to the third Zolotaryov problem in the complex plane, having an approximate closed form solution, and determine the best Cauchy–Hadamard convergence rate. The theory is illustrated with numerical examples for Maxwell's equations arising in 3D magnetotelluric geophysical exploration.

Hardy Space Infinite Elements for Scattering and Resonance Problems

Thorsten Hohage and Lothar Nannen

SIAM J. Numer. Anal. 47, pp. 972-996 (25 pages) | Cited 1 time

Online Publication Date: February 13, 2009

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This paper introduces a new type of infinite element for scattering and resonance problems that is derived from a variant of the pole condition as radiation condition. This condition states that a certain transform of the exterior solution belongs to the Hardy space of $L^2$ boundary values of holomorphic functions on the unit disc if and only if the solution is outgoing. We obtain a symmetric variational formulation of the problem in this Hardy space. Our infinite elements correspond to a Galerkin discretization with respect to the standard monomial orthogonal basis of this Hardy space and lead to simple element matrices. Hardy space infinite elements are particularly well suited for solving resonance problems since they preserve the eigenvalue structure of the problem. We prove superalgebraic convergence for a separated problem. Numerical experiments exhibit fast convergence over a wide range of wave numbers.

Accelerated Line-search and Trust-region Methods

P.-A. Absil and K. A. Gallivan

SIAM J. Numer. Anal. 47, pp. 997-1018 (22 pages)

Online Publication Date: February 13, 2009

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In numerical optimization, line-search and trust-region methods are two important classes of descent schemes, with well-understood global convergence properties. We say that these methods are “accelerated” when the conventional iterate is replaced by any point that produces at least as much of a decrease in the cost function as a fixed fraction of the decrease produced by the conventional iterate. A detailed convergence analysis reveals that global convergence properties of line-search and trust-region methods still hold when the methods are accelerated. The analysis is performed in the general context of optimization on manifolds, of which optimization in $\mathbb{R}^n$ is a particular case. This general convergence analysis sheds new light on the behavior of several existing algorithms.

On Preconditioned Iterative Methods for Certain Time-Dependent Partial Differential Equations

Zhong-Zhi Bai, Yu-Mei Huang, and Michael K. Ng

SIAM J. Numer. Anal. 47, pp. 1019-1037 (19 pages)

Online Publication Date: February 13, 2009

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When the Newton method or the fixed-point method is employed to solve the systems of nonlinear equations arising in the sinc-Galerkin discretization of certain time-dependent partial differential equations, in each iteration step we need to solve a structured subsystem of linear equations iteratively by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the tensor and the Toeplitz structures of the linear subsystems we construct structured preconditioners for their coefficient matrices and estimate the eigenvalue bounds of the preconditioned matrices under certain assumptions. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods. It has been shown that a combination of the Newton/fixed-point iteration with the preconditioned GMRES method is efficient and robust for solving the systems of nonlinear equations arising from the sinc-Galerkin discretization of the time-dependent partial differential equations.

Convergence Analysis of a Discontinuous Galerkin Method with Plane Waves and Lagrange Multipliers for the Solution of Helmholtz Problems

Mohamed Amara, Rabia Djellouli, and Charbel Farhat

SIAM J. Numer. Anal. 47, pp. 1038-1066 (29 pages) | Cited 1 time

Online Publication Date: February 13, 2009

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We analyze the convergence of a discontinuous Galerkin method (DGM) with plane waves and Lagrange multipliers that was recently proposed by Farhat, Harari, and Hetmaniuk [Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 1389–1419] for solving two-dimensional Helmholtz problems at relatively high wavenumbers. We prove that the underlying hybrid variational formulation is well-posed. We also present various a priori error estimates that establish the convergence and order of accuracy of the simplest-1.5pt element associated with this method. We prove that, for $k\,(k\,h)^{\frac{2}{3}}$ sufficiently-1.5pt small, the relative error in the $L^{2}$-norm (resp. in the $H^1$ seminorm) is of order $k\,(k\,h)^{\frac{4}{3}}$ (resp. of order $(k\,h)^{\frac{2}{3}}$) for a solution being in $H^{\frac{5}{3}}(\Omega)$. In addition, we establish an a posteriori error estimate that can be used as a practical error indicator when refining the partition of the computational domain.

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

S. Giani and I. G. Graham

SIAM J. Numer. Anal. 47, pp. 1067-1091 (25 pages) | Cited 2 times

Online Publication Date: February 13, 2009

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We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow

Bernardo Cockburn and Jayadeep Gopalakrishnan

SIAM J. Numer. Anal. 47, pp. 1092-1125 (34 pages)

Online Publication Date: February 19, 2009

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In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

Numerical Analysis of a Finite Element/Volume Penalty Method

Bertrand Maury

SIAM J. Numer. Anal. 47, pp. 1126-1148 (23 pages)

Online Publication Date: February 19, 2009

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We present here some contributions to the numerical analysis of the penalty method in the finite element context. We are especially interested in the ability provided by this approach to use Cartesian, non boundary-fitted meshes to solve elliptic problems in complicated domain. In the spirit of fictitious domains, the initial problem is replaced by a penalized one, posed over a simply shaped domain which covers the original one. This method relies on two parameters, namely $h$ (space-discretization parameter) and $\varepsilon$ (penalty parameter). We propose here a general strategy to estimate the error in both parameters, and we present how it can be applied to various situations. We pay special attention to a scalar version of the rigid motion constraint for fluid-particle flows.

Modified Combined Field Integral Equations for Electromagnetic Scattering

O. Steinbach and M. Windisch

SIAM J. Numer. Anal. 47, pp. 1149-1167 (19 pages) | Cited 1 time

Online Publication Date: February 19, 2009

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The boundary integral formulation of exterior boundary value problems for the Maxwell system may not be equivalent to the original uniquely solvable problem if the wave number corresponds to an eigenvalue of an associated interior eigenvalue problem. To avoid these spurious modes one may use a combined boundary integral approach. To analyze the resulting boundary integral equations in the energy function spaces suitable regularizations have to be introduced. Here we formulate and analyze a modified boundary integral equation which is based on the use of standard boundary integral operators only. A first numerical example shows the applicability of the proposed approach.

A Fast Method for Linear Waves Based on Geometrical Optics

Christiaan C. Stolk

SIAM J. Numer. Anal. 47, pp. 1168-1194 (27 pages) | Cited 1 time

Online Publication Date: February 19, 2009

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We develop a fast method for solving the one-dimensional wave equation based on geometrical optics. From geometrical optics (e.g., Fourier integral operator theory or WKB approximation) it is known that high-frequency waves split into forward and backward propagating parts, each propagating with the wave speed, with amplitude that is slowly changing depending on the medium coefficients, under the assumption that the medium coefficients vary slowly compared to the wavelength. Based on this we construct a method of optimal, $O(N)$ complexity, with basically the following steps: 1. decouple the wavefield into an approximately forward and an approximately backward propagating part; 2. propagate each component explicitly along the characteristics over a time step that is small compared to the medium scale but can be large compared to the wavelength; 3. apply a correction to account for the errors in the explicit propagation; repeat steps 2 and 3 over the necessary amount of time steps; and 4. reconstruct the full field by adding forward and backward propagating components again. Due to step 3 the method accurately computes the full wavefield. A variant of the method was implemented and outperformed a standard order (4,4) finite difference method by a substantial factor. The general principle is applicable also in higher dimensions, but requires efficient implementations of Fourier integral operators which are still the subject of current research.

Stable and Compatible Polynomial Extensions in Three Dimensions and Applications to the $p$ and $h$-$p$ Finite Element Method

Benqi Guo and Jianming Zhang

SIAM J. Numer. Anal. 47, pp. 1195-1225 (31 pages) | Cited 2 times

Online Publication Date: February 25, 2009

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Polynomial extensions play a vital role in the analysis of the $p$ and $h$-$p$ finite element method (FEM) and the spectral element method. We construct explicitly polynomial extensions on standard elements: cubes and triangular prisms, which together with the extension on tetrahedrons are used by the $p$ and $h$-$p$ FEM in three dimensions. These extensions are proved to be stable and compatible with FEM subspaces on tetrahedrons, cubes, and prisms and realize a continuous mapping: $H^{1/2}_{00}(T)$ (or $H^{1/2}_{00}(S))\rightarrow H^1(\Omega_{st})$, where $\Omega_{st}$ denotes one of these standard elements and $T$ and $S$ are their triangular and square faces. Applications of these polynomial extensions to the $p$ and $h$-$p$ FEM are illustrated.

Mixed Finite Element Methods for the Fully Nonlinear Monge–Ampère Equation Based on the Vanishing Moment Method

Xiaobing Feng and Michael Neilan

SIAM J. Numer. Anal. 47, pp. 1226-1250 (25 pages) | Cited 2 times

Online Publication Date: February 25, 2009

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This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge–Ampère equation $\det(D^2u^0)=f\,(>0)$ based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge–Ampère equation is approximated by the fourth order quasilinear equation $-\varepsilon\Delta^2 u^\varepsilon + \det{D^2u^\varepsilon}=f$. It was proved in [X. Feng, Trans. AMS, submitted] that the solution $u^\varepsilon$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge–Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second-order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann–Miyoshi-type mixed finite element methods for approximating the solution $u^\varepsilon$ of the regularized fourth-order problem, which computes simultaneously $u^\varepsilon$ and the moment tensor $\sigma^\varepsilon:=D^2u^\varepsilon$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\varepsilon$, for the errors $u^\varepsilon-u^\varepsilon_h$ and $\sigma^0-\sigma_h^\varepsilon$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\varepsilon$ for the error $u^0-u_h^\varepsilon$ and $\sigma^\varepsilon-\sigma_h^\varepsilon$, and numerically examine what is the “best” mesh size $h$ in relation to $\varepsilon$ in order to achieve these rates. Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations.

Nonsmooth Newton Methods for Set-Valued Saddle Point Problems

Carsten Gräser and Ralf Kornhuber

SIAM J. Numer. Anal. 47, pp. 1251-1273 (23 pages)

Online Publication Date: February 25, 2009

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We present a new class of iterative schemes for large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach.

The Local $L^2$ Projected $C^0$ Finite Element Method for Maxwell Problem

Huo-Yuan Duan, Feng Jia, Ping Lin, and Roger C. E. Tan

SIAM J. Numer. Anal. 47, pp. 1274-1303 (30 pages) | Cited 1 time

Online Publication Date: February 25, 2009

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An element-local $L^2$-projected $C^0$ finite element method is presented to approximate the nonsmooth solution being not in $H^1$ of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local $L^2$ projectors are applied to both curl and div operators. The $C^0$ linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in $L^2$ norm is established uniform in the mesh-size, and the condition number ${\cal O}(h^{-2})$ of the resulting linear system is proven. For the solution and its curl in $H^r$ with $r<1$ we obtain an error bound ${\cal O}(h^r)$ in an energy norm. Numerical experiments confirm the theoretical error bound.

On the Existence of Explicit $hp$-Finite Element Methods Using Gauss–Lobatto Integration on the Triangle

B. T. Helenbrook

SIAM J. Numer. Anal. 47, pp. 1304-1318 (15 pages) | Cited 1 time

Online Publication Date: February 25, 2009

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Spectral-element simulations on quadrilaterals and hexahedra rely on the Gauss–Lobatto (GL) integration rule to enable explicit simulations with optimal spatial convergence rates. In this work, it is proved that a similar integration rule does not exist on triangles. The following properties of the rule are sought: a $(p+1)(p+2)/2$ point integration rule capable of exactly integrating the space given by ${\cal T}(2p -1)\equiv\{x^my^n|0\leq m,n;m+n\leq2p-1\}$, where $p$ is an integer; integration points located at each of the triangle vertices; $p-1$ integration points located on each side; and $(p-1)(p-2)/2$ integration points located in the interior of the element. The proof hinges on the fact that the existence of such a rule implies the existence of a nodal basis with an approximate diagonal mass matrix that can be inverted to obtain exact Galerkin projections of functions in ${\cal T}(p-1)$. The proof shows that vertex functions of a basis having this property exist and are unique, but on a triangle these functions are not nodal, and therefore the GL rule does not exist. In spite of this, the existence of the vertex functions indicates that there may be a nonnodal basis that has the above property. This basis would enable explicit $hp$-finite element simulations on the triangle with optimal spatial accuracy. The methodology developed in the paper gives insight into a possible way to find such a basis.

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov

SIAM J. Numer. Anal. 47, pp. 1319-1365 (47 pages) | Cited 7 times

Online Publication Date: February 25, 2009

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We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, nonconforming, and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric, and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain, which are then automatically coupled. Finally, the framework brings about a new point of view, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.

Numerical Dispersive Schemes for the Nonlinear Schrödinger Equation

Liviu I. Ignat and Enrique Zuazua

SIAM J. Numer. Anal. 47, pp. 1366-1390 (25 pages)

Online Publication Date: February 25, 2009

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We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with $L^2$-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

Blanca Ayuso and L. Donatella Marini

SIAM J. Numer. Anal. 47, pp. 1391-1420 (30 pages) | Cited 2 times

Online Publication Date: February 25, 2009

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We apply the weighted-residual approach recently introduced in [F. Brezzi et al., Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 3293–3310] to derive discontinuous Galerkin formulations for advection-diffusion-reaction problems. We devise the basic ingredients to ensure stability and optimal error estimates in suitable norms, and propose two new methods.

On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces

Qiang Du, Desheng Wang, and Liyong Zhu

SIAM J. Numer. Anal. 47, pp. 1421-1444 (24 pages) | Cited 1 time

Online Publication Date: March 13, 2009

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The performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation (PDE) models. It is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix. In this paper, for typical second-order elliptic problems, some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes. The derivation of such relations for general high-order elements is based on a new trace formula for the element stiffness matrix. It is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces. These results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing.

Dynamical Systems and Non-Hermitian Iterative Eigensolvers

Mark Embree and Richard B. Lehoucq

SIAM J. Numer. Anal. 47, pp. 1445-1473 (29 pages) | Cited 1 time

Online Publication Date: March 13, 2009

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Simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms. We observe that these simple methods can be viewed as forward Euler discretizations of well-known autonomous differential equations that enjoy appealing geometric properties. This connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue, provides insight into the role of orthogonality and biorthogonality, and suggests the development of new methods and analyses based on more sophisticated discretizations. These results also highlight the effect of preconditioning on the convergence and stability of the continuous-time system and its discretization.

A New Fictitious Domain Approach Inspired by the Extended Finite Element Method

Jaroslav Haslinger and Yves Renard

SIAM J. Numer. Anal. 47, pp. 1474-1499 (26 pages)

Online Publication Date: March 25, 2009

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The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moës, Dolbow, and Belytschko in [Internat. J. Numer. Methods Engrg., 46 (1999), pp. 131–150]. An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.

A Saddle Point Approach to the Computation of Harmonic Maps

Qiya Hu, Xue-Cheng Tai, and Ragnar Winther

SIAM J. Numer. Anal. 47, pp. 1500-1523 (24 pages)

Online Publication Date: March 25, 2009

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In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed.

First-Order System Least-Squares Methods for an Optimal Control Problem by the Stokes Flow

Soorok Ryu, Hyung-Chun Lee, and Sang Dong Kim

SIAM J. Numer. Anal. 47, pp. 1524-1545 (22 pages)

Online Publication Date: April 01, 2009

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The least-squares approximations of an optimal control problem governed by the Stokes equations are considered, which leads to an unconstrained coupled optimization problem by the Lagrange multiplier method. The least-squares functionals for the two- and three-dimensional first-order coupled optimality systems are employed by modifying those functionals in [Z. Cai, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 1727–1741]. The established ellipticity and continuity in a product $H^1$ norm yield the optimal discretization error estimates in the finite element spaces. For numerical tests, we apply V-cycle multigrid methods to the whole discrete algebraic system.

Estimating Multidimensional Density Functions Using the Malliavin–Thalmaier Formula

A. Kohatsu-Higa and Kazuhiro Yasuda

SIAM J. Numer. Anal. 47, pp. 1546-1575 (30 pages)

Online Publication Date: April 01, 2009

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The Malliavin–Thalmaier formula was introduced in [P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance, Springer-Verlag, Berlin, 2006] as an alternative expression for the density of a multivariate smooth random variable in Wiener space. In comparison with classical integration by parts formulae, this alternative formulation requires the application of the integration by parts formula only once to obtain an expression that can be simulated. Therefore, this expression is free from the curse of dimensionality. Unfortunately, when this formula is applied directly in computer simulation, it exhibits unstable behavior. We propose an approximation to the Malliavin–Thalmaier formula in the spirit of the theory of kernel density estimation to solve this problem. In the first part of this paper, we obtain a central limit theorem for the estimation error. And in the latter part, we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance.

A Three-Level BDDC Algorithm for Mortar Discretizations

Hyea Hyun Kim and Xuemin Tu

SIAM J. Numer. Anal. 47, pp. 1576-1600 (25 pages) | Cited 1 time

Online Publication Date: April 16, 2009

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In this paper, a three-level balancing domain decomposition by constraints (BDDC) algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically nonconforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work: the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method, and numerical experiments are also discussed.
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  1. An Optimal Adaptive Finite Element Method for the Stokes Problem
  2. Adaptive Multiprecision Path Tracking
  3. New Methods of Controlled Total Variation Reduction for Digital Functions
  4. Nonclassical Shocks and Kinetic Relations: Finite Difference Schemes
  5. High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation L...
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