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SIAM J. on Numerical Analysis

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2008

Volume 46, Issue 2, pp. 545-1095


Convergence Analysis for The Numerical Boundary Corrector for Elliptic Equations with Rapidly Oscillating Coefficients

Marcus Sarkis and Henrique Versieux

SIAM J. Numer. Anal. 46, pp. 545-576 (32 pages)

Online Publication Date: February 01, 2008

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We develop the convergence analysis of a numerical scheme for approximating the solution of the elliptic problem $L_{\epsilon}u_{\epsilon} =- \frac{\partial}{\partial x_{i}}a_{ij}(x/ \epsilon) \frac{\partial }{\partial x_{j}}u_{\epsilon}=f \mbox{in} \Omega, u_{\epsilon}=0 \mbox{on} \partial\Omega,$ where $a(y)=(a_{ij}(y))$ is a periodic symmetric positive definite matrix and $\Omega = (0,1)^2$. The major goal of the numerical scheme is to capture the $\epsilon$-scale of the oscillations of the solution $u_\epsilon$ on a mesh size $h>\epsilon (\mbox{or} h>>\epsilon)$. The numerical scheme is based on asymptotic expansions, constructive boundary corrector, and finite element approximations. New a priori error estimates are established for the asymptotic expansions and for the constructive boundary correctors under weak assumptions on the regularity of the problem. These estimates permit to establish sharp finite element error estimates and to consider composite materials applications. Depending on the regularity of the problem, we establish for the numerical scheme a priori error estimates of $O(h^2 + \epsilon^{3/2}+ \epsilon h )$ on the $L^2$-norm, and $O(h + \epsilon^{1+ \hat{\delta}})$ for the broken $H^1$-norm where $\hat{\delta} \in (-\frac{1}{4}, 0]$.

Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints

Massimo Fornasier and Holger Rauhut

SIAM J. Numer. Anal. 46, pp. 577-613 (37 pages) | Cited 8 times

Online Publication Date: February 01, 2008

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Vector-valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of nonvanishing components. These measures are typically constructed as weighted $\ell_1$ norms of componentwise $\ell_q$ norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its $\ell_2$-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.

Linear Convergence of an Adaptive Finite Element Method for the $p$-Laplacian Equation

Lars Diening and Christian Kreuzer

SIAM J. Numer. Anal. 46, pp. 614-638 (25 pages) | Cited 4 times

Online Publication Date: February 01, 2008

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We study an adaptive finite element method for the $p$-Laplacian like PDEs using piecewise linear, continuous functions. The error is measured by means of the quasi norm of Barrett and Liu. We provide residual based error estimators without a gap between the upper and lower bound. We show linear convergence of the algorithm which is similar to the one of Morin, Nochetto, and Siebert. All results are obtained without extra marking for the oscillation.

Convergence of a High-Order Semi-Lagrangian Scheme with Propagation of Gradients for the One-Dimensional Vlasov–Poisson System

Nicolas Besse

SIAM J. Numer. Anal. 46, pp. 639-670 (32 pages) | Cited 1 time

Online Publication Date: February 01, 2008

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In this paper we give a proof of convergence of a new numerical method introduced in [N. Besse and E. Sonnendrücker, J. Comput. Phys., 191 (2003), pp. 341–376] for the Vlasov equation. The numerical method is based on the semi-Lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high-order and stable reconstruction. These kinds of new schemes have been successfully implemented on unstructured meshes of four-dimensional phase space (cf. [N. Besse, Etude mathématique et numérique de l'equation de Vlasov sur des maillages non structurés de l'espace des phases, thèse de l'Université Louis Pasteur, Strasbourg, France, 2003; N. Besse, J. Segré, and E. Sonnendrücker, Transport Theory Statist. Phys., 34 (2005), pp. 311–332]). In order to make a rigorous proof of convergence of this method and simplify the convergence analysis, we have considered the periodic one-dimensional Vlasov–Poisson system in phase space on a grid. The distribution $f(t,x,v)$ and the electric field are shown to converge to the exact solution values in $H^1$ norm. The rate of convergence is of ${\mathcal{O}(\Delta t^2 +\frac{\Delta x^{4-|\alpha|}}{\Delta t}+ \frac{\Delta v^{4-|\alpha|}} {\Delta t})}$, $\alpha \in \N^2, |\alpha|\leq 1$.

On Sinc-Based Method in Computing Eigenvalues of Boundary-Value Problems

M. H. Annaby and R. M. Asharabi

SIAM J. Numer. Anal. 46, pp. 671-690 (20 pages)

Online Publication Date: February 01, 2008

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We aim in this paper to give a complete treatment of the error analysis of the use of the sinc method in computing eigenvalues of second order Birkhoff-regular eigenvalue problems. For this task we study truncation and amplitude errors on $\Bbb C$ associated with cardinal series representation of band-limited functions as well as those of their derivatives. These results, which seem to be considered for the first time, give us the possibility to establish the error analysis when the eigenvalues are not real or not algebraically simple.

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Adrien Blanchet, Vincent Calvez, and José A. Carrillo

SIAM J. Numer. Anal. 46, pp. 691-721 (31 pages) | Cited 1 time

Online Publication Date: February 01, 2008

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Variational steepest descent approximation schemes for the modified Patlak–Keller–Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated with this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weak solutions to the modified Patlak–Keller–Segel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in dimension one. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce the blow-up of solutions for supercritical masses easily without the need of mesh-refinement.

Adaptive Multiprecision Path Tracking

Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, II

SIAM J. Numer. Anal. 46, pp. 722-746 (25 pages) | Cited 2 times

Online Publication Date: February 08, 2008

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This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

An Optimal Adaptive Finite Element Method for the Stokes Problem

Yaroslav Kondratyuk and Rob Stevenson

SIAM J. Numer. Anal. 46, pp. 747-775 (29 pages) | Cited 4 times

Online Publication Date: February 08, 2008

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A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. The method consists of 3 nested loops. The outermost loop consists of an adaptive finite element method for solving the pressure from the (elliptic) Schur complement system that arises by eliminating the velocity. Each of the arising finite element problems is a Stokes-type problem, with the pressure being sought in the current trial space and the divergence-free constraint being reduced to orthogonality of the divergence to this trial space. Such a problem is still continuous in the velocity field. In the middle loop, its solution is approximated using the Uzawa scheme. In the innermost loop, the solution of the elliptic system for the velocity field that has to be solved in each Uzawa iteration is approximated by an adaptive finite element method.

Discontinuous Galerkin Methods for Friedrichs' Systems. Part III. Multifield Theories with Partial Coercivity

Alexandre Ern and Jean-Luc Guermond

SIAM J. Numer. Anal. 46, pp. 776-804 (29 pages) | Cited 2 times

Online Publication Date: February 08, 2008

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This paper is the third and last part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The purpose of this paper is to extend the framework that has been developed in Part II for two-field Friedrichs' systems associated with second-order PDEs. We now consider two-field Friedrichs' systems with partial $L^2$-coercivity and three-field Friedrichs' systems with an even weaker $L^2$-coercivity hypothesis. In particular, this work generalizes the discontinuous Galerkin methods of Part II to compressible and incompressible linear continuum mechanics. We also show how the stabilizing parameters of the method must be set when the two-field Friedrichs' system is composed of terms that may be of different magnitude, thus accounting, for instance, for advection–diffusion equations at high Péclet numbers.

Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection

Daniele A. Di Pietro, Alexandre Ern, and Jean-Luc Guermond

SIAM J. Numer. Anal. 46, pp. 805-831 (27 pages) | Cited 2 times

Online Publication Date: February 08, 2008

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We construct and analyze a discontinuous Galerkin method to solve advection-diffusion-reaction PDEs with anisotropic and semidefinite diffusion. The method is designed to automatically detect the so-called elliptic/hyperbolic interface on fitted meshes. The key idea is to use consistent weighted average and jump operators. Optimal estimates in the broken graph norm are proven. These are consistent with well-known results when the problem is either hyperbolic or uniformly elliptic. The theoretical results are supported by numerical evidence.

Local Regularization for the Nonlinear Inverse Autoconvolution Problem

Zhewei Dai and Patricia K. Lamm

SIAM J. Numer. Anal. 46, pp. 832-868 (37 pages)

Online Publication Date: February 20, 2008

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We develop a local regularization theory for the nonlinear inverse autoconvolution problem. Unlike classical regularization techniques such as Tikhonov regularization, this theory provides regularization methods that preserve the causal nature of the autoconvolution problem, allowing for fast sequential numerical solution (${\cal O}(rN^2 - r^2N)$ flops, where $r\ll N$ for the method discussed in this paper as applied to the nonlinear problem; in comparison, the cost for Tikhonov regularization applied to a general linear problem is ${\cal O}(N^3)$ flops). We prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both $L_2$ and continuous data. We propose several regularization methods and provide a theoretical basis for their convergence; of note is that this class of methods does not require an initial guess of the unknown solution. Our numerical results confirm the effectiveness of the methods, with results comparing favorably to numerical examples found in the literature for the autoconvolution problem (e.g., [G. Fleischer, R. Gorenflo, and B. Hofmann, ZAMM Z. Angew. Math. Mech., 79 (1999), pp. 149–159] for examples using Tikhonov regularization with total variation constraints and [J. Janno, Inverse Problems, 16 (2000), pp. 333–348] for examples using the method of Lavrent'ev); this especially seems to be true when it comes to the recovery of sharp features in the unknown solution. We also show the effectiveness of our method in cases not covered by the theory.

Asymptotic High-Order Schemes for $2\times2$ Dissipative Hyperbolic Systems

Denise Aregba-Driollet, Maya Briani, and Roberto Natalini

SIAM J. Numer. Anal. 46, pp. 869-894 (26 pages) | Cited 2 times

Online Publication Date: February 20, 2008

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We investigate finite difference schemes which approximate $2\times 2$ one-dimensional linear dissipative hyperbolic systems. We show that it is possible to introduce some suitable modifications in standard upwinding schemes, which keep into account the long-time behavior of the solutions, to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states, respectively, around stationary solutions and in the diffusion (Chapman–Enskog) limit.

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

V. M. Devigne, I. S. Pop, C. J. van Duijn, and T. Clopeau

SIAM J. Numer. Anal. 46, pp. 895-919 (25 pages)

Online Publication Date: February 22, 2008

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In this paper we analyze a numerical scheme for a dissolution and precipitation model in porous media. We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary. We use a regularization approach for constructing a semi-implicit scheme that is stable and convergent. For dealing with the emerging time discrete nonlinear problems, we propose a simple fixed-point iterative procedure. The paper is concluded by numerical results.

Convexity-Preserving Piecewise Rational Quartic Interpolation

Xuli Han

SIAM J. Numer. Anal. 46, pp. 920-929 (10 pages)

Online Publication Date: March 05, 2008

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A family of piecewise rational quartic interpolants is given. Identified uniquely by the value of a tension parameter $\lambda_i$, each interpolant of the family can be $C^2$ spline without solving a linear or nonlinear system of consistency equations for the derivative values at the knots. The interpolant can preserve the local convexity/concavity properties of the given data. A proper choice of $\lambda_i$ to guarantee shape preservation is given. A convergence analysis establishes an error bound in terms of $\lambda_i$ and shows that $O(h^3)$ accuracy is obtained for $C^2$ continuity. Several examples are supplied to support the practical value of the method.

New Quadrature Formulas from Conformal Maps

Nicholas Hale and Lloyd N. Trefethen

SIAM J. Numer. Anal. 46, pp. 930-948 (19 pages) | Cited 4 times

Online Publication Date: March 05, 2008

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Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of $\pi/2$ with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight-sided domain. The new methods are compared with related ideas of Bakhvalov, Kosloff and Tal-Ezer, Rokhlin and Alpert, and others. An advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands. For example, one of the formulas presented is proved to converge $50\%$ faster than Gauss quadrature for functions analytic in an $\varepsilon$-neighborhood of $[-1,1]$.

Coupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements

Vivette Girault, Shuyu Sun, Mary F. Wheeler, and Ivan Yotov

SIAM J. Numer. Anal. 46, pp. 949-979 (31 pages) | Cited 1 time

Online Publication Date: March 05, 2008

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Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale $h$ and the coarse mortar scale $H$ are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed.

Stability Analysis of the Gurtin–MacCamy Model

D. Breda, M. Iannelli, S. Maset, and R. Vermiglio

SIAM J. Numer. Anal. 46, pp. 980-995 (16 pages)

Online Publication Date: March 05, 2008

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In this paper we propose a numerical scheme to investigate the stability of steady states of the nonlinear Gurtin–MacCamy system, which is a basic model in population dynamics. In fact the analysis of stability is usually performed by the study of transcendental characteristic equations that are too difficult to approach by analytical methods. The method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques. The method computes the rightmost characteristic roots, and it is shown to converge with spectral accuracy behavior.

On Stability of Staggered Schemes

Amy L. Bauer, Raphaël Loubère, and Burton Wendroff

SIAM J. Numer. Anal. 46, pp. 996-1011 (16 pages)

Online Publication Date: March 05, 2008

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This paper investigates the theoretical stability bound of a Lagrangian staggered scheme used to solve hydrodynamics equations. We present the two-dimensional (2D) wave equation as a possible model for this study and, by using the numerical radius of the amplification matrix, we prove that the family of schemes defined with two time-centering parameters is limited by a nonclassical stability bound limit defined with an analytical curve. We further show that 2D numerical experiments agree with this theoretical result.

Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws

Manuel Castro, José M. Gallardo, Juan A. López-GarcÍa, and Carlos Parés

SIAM J. Numer. Anal. 46, pp. 1012-1039 (28 pages)

Online Publication Date: March 05, 2008

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This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.

Mollified Impulse Methods for Highly Oscillatory Differential Equations

J. M. Sanz-Serna

SIAM J. Numer. Anal. 46, pp. 1040-1059 (20 pages) | Cited 3 times

Online Publication Date: March 05, 2008

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We introduce a family of impulselike methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast part and a slow part. Methods of this family are specified by two weight functions $\phi$, $\psi$; one is used to average positions and the other to mollify the force. When the fast forces are conservative and $\phi=\psi$, the methods here coincide with the mollified impulse methods introduced by García-Archilla, Sanz-Serna, and Skeel. On the other hand, the methods here extend to nonlinear situations a well-known class of exponential integrators introduced by Hairer and Lubich for cases of linear fast forces. A convergence analysis is presented that provides insight into the role played by the processes of mollification and averaging in avoiding order reduction. A simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction.

A Spectral Viscosity Method Based on Hermite Functions for Nonlinear Conservation Laws

Julián Aguirre and Judith Rivas

SIAM J. Numer. Anal. 46, pp. 1060-1078 (19 pages) | Cited 1 time

Online Publication Date: March 05, 2008

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We consider the approximation by a spectral method of the solution of the Cauchy problem for a scalar conservation law in one dimension posed in the whole real line. We analyze a spectral viscosity method in which the orthogonal basis considered is the one of Hermite functions. We prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments.

New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation

P. Zhuang, F. Liu, V. Anh, and I. Turner

SIAM J. Numer. Anal. 46, pp. 1079-1095 (17 pages) | Cited 4 times

Online Publication Date: March 05, 2008

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A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.
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