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SIAM J. on Numerical Analysis

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2011

Volume 49, Issue 6, pp. 2211-2639


An Error Corrected Euler Method for Solving Stiff Problems Based on Chebyshev Collocation

Philsu Kim, Xiangfan Piao, and Sang Dong Kim

SIAM J. Numer. Anal. 49, pp. 2211-2230 (20 pages)

Online Publication Date: November 01, 2011

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In this paper, we present error corrected Euler methods for solving stiff initial value problems, which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess. The proposed methods use a Chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler's polygon. These methods with or without the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences.

Fast Multilevel Augmentation Methods for Nonlinear Boundary Integral Equations

Xiangling Chen, Zhongying Chen, Bin Wu, and Yuesheng Xu

SIAM J. Numer. Anal. 49, pp. 2231-2255 (25 pages)

Online Publication Date: November 01, 2011

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We propose a fast algorithm for the solution of the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. The fast algorithm is developed by using the multilevel augmentation method (introduced recently by Chen, Wu, and Xu for general nonlinear integral equations), in conjunction with a matrix truncation strategy, and an error control technique of numerical integrations for integrals appeared in the process of solving the equation. We prove that the proposed algorithm has an optimal convergence order (up to a logarithmic factor) and a nearly linear computational complexity order (measured in the number of multiplications and functional evaluations). Numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency, verifying the theoretical estimates, and to compare performance of the proposed algorithm with that of the Atkinson and Chandler algorithm.

Maximum-Norm Estimates for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms

Timothy A. Lucas

SIAM J. Numer. Anal. 49, pp. 2256-2276 (21 pages)

Online Publication Date: November 01, 2011

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This paper describes numerical methods and the corresponding maximum-norm error estimates for a chemotaxis model proposed by Kepler. Upon detecting pathogens, immune cells secrete soluble factors that attract other immune cells to the site of the infection. The motion of the model cells is stochastic, but biased toward the gradient of one or more of the soluble factors. The soluble factors are modeled by a system of reaction-diffusion equations with sources that are centered on the cells. Previously, I presented a first order splitting in time for solving the reaction-diffusion-stochastic system numerically. The diffusion, reaction, and stochastic differential equations can be approximated separately to first order in the supremum norm. The three-dimensional domain is discretized using finite elements, and the diffusion is solved using a backward Euler scheme combined with multigrid. The reaction is solved using a simple semi-implicit first order scheme. The stochastic differential equations are given by a Langevin process which can be simulated exactly. The paper concludes by demonstrating first order convergence of the entire simulation and providing a sample simulation of the reaction-diffusion-stochastic system.

A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.

Marcel Bieri

SIAM J. Numer. Anal. 49, pp. 2277-2301 (25 pages)

Online Publication Date: November 08, 2011

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This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.

Error Estimates of Crank–Nicolson-Type Difference Schemes for the Subdiffusion Equation

Ya-nan Zhang, Zhi-zhong Sun, and Hong-wei Wu

SIAM J. Numer. Anal. 49, pp. 2302-2322 (21 pages)

Online Publication Date: November 08, 2011

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A Crank–Nicolson-type difference scheme is proposed for solving the subdiffusion equation with fractional derivative, and the truncation error is analyzed in detail. At each temporal level, only a tridiagonal linear system needs to be solved and the Thomas algorithm may be used. The solvability, unconditional stability, and $H^1$ norm convergence are proved. The convergence order is ${\rm min}\{2-{\gamma}/{2},\;1+\gamma\}$ in the temporal direction and two in the spatial direction. By the Sobolev embedding inequality, we obtain the maximum norm error estimate. A spatial compact scheme based on the Crank–Nicolson-type difference scheme is also presented, and similar results are given. The convergence order is $\mathcal{O}(\tau^{{\rm min}\{2-{\gamma}/{2},\;1+\gamma\}}+h^4)$. Numerical experiments are included to support the theoretical results, and comparisons with the related works are presented to show the effectiveness of our method.

Mixed Finite Element Methods of Higher-Order for Model Contact Problems

Andreas Schröder

SIAM J. Numer. Anal. 49, pp. 2323-2339 (17 pages)

Online Publication Date: November 08, 2011

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This paper presents mixed finite element methods of higher-order for a simplified Signorini problem and an idealized frictional problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al., which is extended to higher-order finite elements. To guarantee the unique existence of the solution of the mixed method, a discrete inf-sup condition is proven. Approximation results of the $p$-method of finite elements and some inverse estimates for higher-order polynomials are applied. Numerical results confirm the theoretical findings.

Wavenumber-Explicit $hp$-BEM for High Frequency Scattering

Maike Löhndorf and Jens Markus Melenk

SIAM J. Numer. Anal. 49, pp. 2340-2363 (24 pages)

Online Publication Date: November 10, 2011

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For the Helmholtz equation (with wavenumber $k$) and analytic curves or surfaces $\Gamma$, we analyze the Galerkin discretization of classical combined field integral equations in an $L^2$-setting. We give abstract conditions on the approximation properties of the ansatz space that ensure stability and quasi-optimality of the Galerkin method. Special attention is paid to the $hp$-version of the boundary element method ($hp$-BEM). Under the assumption of polynomial growth of the solution operator we show stability and quasi-optimality of the $hp$-BEM if the following scale resolution condition is satisfied: the polynomial degree $p$ is at least $O(\log k)$ and $kh/p$ is bounded by a number that is sufficiently small, but independent of $k$. Under this assumption, the constant in the quasi-optimality estimate is independent of $k$. Numerical examples in two dimensions illustrate the theoretical results and even suggest that in many cases quasi-optimality is given under the weaker condition that $kh/p$ is sufficiently small.

Guaranteed A Posteriori Error Estimator for Mixed Finite Element Methods of Linear Elasticity with Weak Stress Symmetry

Kwang-Yeon Kim

SIAM J. Numer. Anal. 49, pp. 2364-2385 (22 pages)

Online Publication Date: November 22, 2011

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In this paper we propose an a posteriori error estimator for the mixed finite element methods of the linear elasticity problem with the symmetry condition weakly imposed on the stress tensor. The error estimator is constructed by making a proper decomposition of the stress error and using an argument similar to the hypercircle method. It is shown that the resulting estimator yields a guaranteed upper bound on the stress error which relies on computable upper bounds of the constants in the first and second Korn inequalities. We also establish the local lower bound by using the discrete Friedrichs inequality. Our approach is equivalent to the Helmholtz decomposition of the stress error but requires assumptions neither on the regularity of the solution nor the geometry of the domain. Numerical results are provided to illustrate the efficiency of our error estimator.

Heuristic Modified Equation Analysis on Oscillations in Numerical Solutions of Conservation Laws

Jiequan Li and Zhicheng Yang

SIAM J. Numer. Anal. 49, pp. 2386-2406 (21 pages)

Online Publication Date: November 22, 2011

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Oscillations are ubiquitous in numerical solutions obtained by high order or even first order schemes for hyperbolic problems and are conventionally understood as the consequence of low dissipation effects of underlying numerical schemes. Earlier analysis was done mainly through the effective discrete Fourier analysis for linear problems or the modified equation approach in smooth solution regions. In this paper, a so-called heuristic modified equation is derived when applied to nonlinear problems, particularly for oscillatory modes of solutions whose counterpart in linear problems are high frequency mode solutions, and the dissipation effect is distinguished as a numerical damping and a numerical diffusion. The former is reflected through the zero order term of the heuristic modified equation and the latter through the second order differential term. It turns out that the effect of dissipation is categorized as a damping, a neutrality, and an amplification, and that the numerical damping plays a dominant role in offsetting the oscillatory modes. When the amplification effect is taken, the numerical scheme often comes unstable.

Finite Element Approximation of the Cahn–Hilliard–Cook Equation

Mihály Kovács, Stig Larsson, and Ali Mesforush

SIAM J. Numer. Anal. 49, pp. 2407-2429 (23 pages)

Online Publication Date: November 22, 2011

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We study the nonlinear stochastic Cahn–Hilliard equation perturbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to $1$. We also prove strong convergence without known rate.

An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation

Bernhard Beckermann

SIAM J. Numer. Anal. 49, pp. 2430-2450 (21 pages)

Online Publication Date: November 22, 2011

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In this paper we suggest a new formula for the residual of Galerkin projection onto rational Krylov spaces applied to a Sylvester equation, and establish a relation to three different underlying extremal problems for rational functions. These extremal problems enable us to compare the size of the residual for the above method with that obtained by ADI. In addition, we deduce several new a priori error estimates for Galerkin projection onto rational Krylov spaces, both for the Sylvester and for the Lyapunov equation.

Weighted Marking for Goal-oriented Adaptive Finite Element Methods

Roland Becker, Elodie Estecahandy, and David Trujillo

SIAM J. Numer. Anal. 49, pp. 2451-2469 (19 pages)

Online Publication Date: December 08, 2011

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We propose a new goal-oriented adaptive finite element method, which is based on the weighting of the residuals from the primal and dual problem at each step of the iteration. Our main result is the quasi-optimality of the resulting algorithm. Numerical experiments are reported, showing the convergence behavior of the algorithm.

A Generalized Fast Marching Method for Dislocation Dynamics

Elisabetta Carlini, Nicolas Forcadel, and Régis Monneau

SIAM J. Numer. Anal. 49, pp. 2470-2500 (31 pages)

Online Publication Date: December 15, 2011

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In this paper, we consider a generalized fast marching method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hypersurface in $\mathbb{R}^N$ (with $N=2$ for physical applications) is given by its normal velocity which is a nonlocal function of the whole shape of the hypersurface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension $N=2$.

A Priori and A Posteriori Pseudostress-velocity Mixed Finite Element Error Analysis for the Stokes Problem

Carsten Carstensen, Dongho Kim, and Eun-Jae Park

SIAM J. Numer. Anal. 49, pp. 2501-2523 (23 pages)

Online Publication Date: December 15, 2011

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The pseudostress-velocity formulation of the stationary Stokes problem allows a Raviart–Thomas mixed finite element formulation with quasi-optimal convergence and some superconvergent reconstruction of the velocity. This local postprocessing gives rise to some averaging a posteriori error estimator with explicit constants for reliable error control. Standard residual-based explicit a posteriori error estimation is shown to be reliable and efficient and motivates adaptive mesh-refining algorithms. Numerical experiments confirm our theoretical findings and illustrate the accuracy of the guaranteed upper error bounds even with reduced regularity.

Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem

M. Dashti and A. M. Stuart

SIAM J. Numer. Anal. 49, pp. 2524-2542 (19 pages)

Online Publication Date: December 15, 2011

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We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension $d \le 3$, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen–Loève expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen–Loève expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipschitz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finite-dimensional Karhunen–Loève truncation.

Hard Thresholding Pursuit: An Algorithm for Compressive Sensing

Simon Foucart

SIAM J. Numer. Anal. 49, pp. 2543-2563 (21 pages)

Online Publication Date: December 15, 2011

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We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and the Compressive Sampling Matching Pursuit algorithm, is called Hard Thresholding Pursuit. We study its general convergence and notice in particular that only a finite number of iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the linear system, the Hard Thresholding Pursuit algorithm indeed finds all $s$-sparse solutions. This condition, which reads $\delta_{3 s} < 1/\sqrt{3}$, is heuristically better than the sufficient conditions currently available for other compressive sensing algorithms. It applies to fast versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed under the condition $\delta_{3 s} < 1/\sqrt{3}$. We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.

Goal-Oriented Local A Posteriori Error Estimators for H(div) Least-Squares Finite Element Methods

Zhiqiang Cai and JaEun Ku

SIAM J. Numer. Anal. 49, pp. 2564-2575 (12 pages)

Online Publication Date: December 15, 2011

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We propose a goal-oriented, local a posteriori error estimator for H(div) least-squares (LS) finite element methods. Our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest $D \subset \Omega$. The estimator is obtained from the LS functional by scaling residuals with proper weight coefficients. The weight coefficients are given in terms of local mesh size $h_T$ and a function $\omega_D$ depending on the distance to $D$. This new error estimator measures the pollution effect from the outside region of $D$ and provides a basis for local refinement in order to efficiently approximate the solution in $D$. Numerical experiments show superior performances of our goal-oriented a posteriori estimators over the standard LS functional and global error estimators.

Fast and Robust Numerical Solution of the Richards Equation in Homogeneous Soil

Heiko Berninger, Ralf Kornhuber, and Oliver Sander

SIAM J. Numer. Anal. 49, pp. 2576-2597 (22 pages)

Online Publication Date: December 22, 2011

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We derive and analyze a solver-friendly finite element discretization of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with nonstandard quadrature points. Our approach allows for nonlinear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the nondegenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formulated as discrete convex minimization problems and, therefore, can be solved efficiently by monotone multigrid methods. In numerical examples for two and three space dimensions we observe $L^2$-convergence rates of order $\mathcal{O}(h^2)$ and $H^1$-convergence rates of order $\mathcal{O}(h)$ as well as robust convergence behavior of the multigrid method with respect to extreme choices of soil parameters.

A Second-order Finite Difference Method for Option Pricing Under Jump-diffusion Models

YongHoon Kwon and Younhee Lee

SIAM J. Numer. Anal. 49, pp. 2598-2617 (20 pages)

Online Publication Date: December 22, 2011

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We develop a finite difference method to solve partial integro-differential equations which describe the behavior of option prices under jump-diffusion models. With localization to a bounded domain of the spatial variable, these equations are discretized on uniform grid points over a finite domain of time and spatial variables. The proposed method is based on three time levels and leads to linear systems with tridiagonal matrices. In this paper the stability of the proposed method and the second-order convergence rate with respect to a discrete $\ell^{2}$-norm are proved. Numerical results obtained with European put options under the Merton and Kou models show the behaviors of the stability and the second-order convergence rate.

Strong Stability Preserving Two-step Runge–Kutta Methods

David I. Ketcheson, Sigal Gottlieb, and Colin B. Macdonald

SIAM J. Numer. Anal. 49, pp. 2618-2639 (22 pages)

Online Publication Date: December 22, 2011

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We investigate the strong stability preserving (SSP) property of two-step Runge–Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present explicit TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order weighted essentially non-oscillatory discretizations.
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