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SIAM J. on Numerical Analysis

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1997

Volume 34, Issue 6, pp. 2087-2502


Order Conditions for General Two-Step Runge--Kutta Methods

Ernst Hairer and Gerhard Wanner

SIAM J. Numer. Anal. 34, pp. 2087-2089 (3 pages) | Cited 11 times

Online Publication Date: July 25, 2006

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The use of B-series for the derivation of order conditions for general two-step Runge--Kutta methods is illustrated.

Parallel Domain Decomposition Solver for Adaptive hp Finite Element Methods

J. T. Oden, Abani Patra, and Yusheng Feng

SIAM J. Numer. Anal. 34, pp. 2090-2118 (29 pages) | Cited 4 times

Online Publication Date: July 25, 2006

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In this paper, the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed. Two-level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator. The condition number of the preconditioned system, for Poisson problems in two space dimensions, is proved to be bounded by C(1 + Hp/h)2(1 + log p)2 and Cp(1 + Hp/h)2(1 + log p)2 for different choices of coarse grid operators, where H is the subdomain size, p is the maximum spectral order, h is the size of the smallest element in the subdomain, and C is a constant independent of the mesh parameters. The work here extends the work of Bramble et al. [Math Comp., 47 (1986), pp. 103--134] on the h-version and Babuska et al. [SIAM J. Numer. Anal., 29 (1991), pp. 624--661] on the p-version of the finite element method. A preliminary version of this solver was first announced by Oden, Patra, and Feng in [Domain Decomposition Solver for Adaptive hp Finite Elements, VII Conference on Domain Decomposition, State College, PA, October 1993]. Numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations.

Implementation of Diagonally Implicit Multistage Integration Methods for Ordinary Differential Equations

J. C. Butcher and Z. Jackiewicz

SIAM J. Numer. Anal. 34, pp. 2119-2141 (23 pages) | Cited 7 times

Online Publication Date: July 25, 2006

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We investigate the implementation of diagonally implicit multistage integration methods (DIMSIMs). The implementation issues addressed are the local error estimation, changing stepsize using the Nordsieck technique, and the construction of continuous interpolants. Numerical experiments with a method of order three indicate that the error estimates that have been constructed are very reliable in both a fixed and a variable stepsize environment.

Numerical Methods in the Weak Sense for Stochastic Differential Equations with Small Noise

G. N. Milstein and M. V. Tret'yakov

SIAM J. Numer. Anal. 34, pp. 2142-2167 (26 pages)

Online Publication Date: July 25, 2006

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We propose a new approach to constructing weak numerical methods for finding solutions to stochastic systems with small noise. For these methods we prove an error estimate in terms of products $h^i\varepsilon ^j$ ($h$ is a time increment, $\varepsilon $ is a small parameter). We derive various efficient weak schemes for systems with small noise and study the Talay--Tubaro expansion of their global error. An efficient approach to reducing the Monte-Carlo error is presented. Some of the proposed methods are tested by calculating the Lyapunov exponent of a linear system with small noise.

Relaxation Schemes for Nonlinear Kinetic Equations

E. Gabetta, L. Pareschi, and G. Toscani

SIAM J. Numer. Anal. 34, pp. 2168-2194 (27 pages) | Cited 16 times

Online Publication Date: July 25, 2006

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A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild's approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the main physical properties: positivity, conservation of mass, momentum, and energy. Moreover, for some particular models, the entropy property is also shown to hold.

Covolume Solutions of Three-Dimensional Div-Curl Equations

Roy A. Nicolaides and Xiaonan Wu

SIAM J. Numer. Anal. 34, pp. 2195-2203 (9 pages) | Cited 21 times

Online Publication Date: July 25, 2006

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The Voronoi--Delaunay mesh systems provide a generalization of the classical rectangular staggered meshes to unstructured meshes. In this work, it is shown how such "covolume" discretizations may be applied to div-curl systems in three dimensions. Error estimates are proved and confirmed by a numerical illustration.

On Order Conditions for Partitioned Symplectic Methods

Ander Murua

SIAM J. Numer. Anal. 34, pp. 2204-2211 (8 pages) | Cited 4 times

Online Publication Date: July 25, 2006

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We are concerned with symplectic methods for integrating Hamiltonian systems. We focus our attention on the independent order conditions for symplectic integrators that can be expanded as P-series. This class of methods includes the important family of partitioned Runge--Kutta methods. It is known that, as in the nonpartitioned case, the conditions for a partitioned method to be symplectic act as simplifying assumptions, introducing many redundancies in the order conditions. We show that there is a one-to-one correspondence between the set of independent order conditions for symplectic partitioned methods and a suitable set of oriented graphs that we call H-trees. We count the number of such H-trees, i.e., the number of independent order conditions.

Multiwavelets for Second-Kind Integral Equations

Tobias von Petersdorff, Christoph Schwab, and Reinhold Schneider

SIAM J. Numer. Anal. 34, pp. 2212-2227 (16 pages) | Cited 25 times

Online Publication Date: July 25, 2006

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We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.

Fully Discrete Finite Element Analysis of Multiphase Flow in Groundwater Hydrology

Zhangxin Chen and Richard E. Ewing

SIAM J. Numer. Anal. 34, pp. 2228-2253 (26 pages) | Cited 13 times

Online Publication Date: July 25, 2006

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This paper deals with the development and analysis of a fully discrete finite element method for a nonlinear differential system for describing an air-water system in groundwater hydrology. The nonlinear system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. The saturation equation is approximated by a finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty$-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for nonsevere degeneracy. Implementation of the fractional flow formulation with various nonhomogeneous boundary conditions is also discussed. Results of numerical experiments using the present approach for modeling groundwater flow in porous media are reported.

Convergence of a Multigrid Method for Elliptic Equations with Highly Oscillatory Coefficients

Bjorn Engquist and Erding Luo

SIAM J. Numer. Anal. 34, pp. 2254-2273 (20 pages) | Cited 5 times

Online Publication Date: July 25, 2006

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Standard multigrid methods are not so effective for equationswith highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.

Spline Collocation Differentiation Matrices

Robert D. Russell and Weiwei Sun

SIAM J. Numer. Anal. 34, pp. 2274-2287 (14 pages) | Cited 4 times

Online Publication Date: July 25, 2006

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Spline collocation differentiation matrices arising when solving PDEs with periodic boundary conditions have block circulant structure. In this paper, we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices. We use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic PDEs.

Numerical Solution of the Generalized Airfoil Equation for an Airfoil with a Flap

G. Monegato and I. H. Sloan

SIAM J. Numer. Anal. 34, pp. 2288-2305 (18 pages) | Cited 6 times

Online Publication Date: July 25, 2006

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In this paper we are concerned with the generalized airfoil equation, which governs the pressure across an airfoil oscillating in a wind tunnel. In particular, we examine the case of an airfoil which contains a flap, so that the input function has a simple jump at a point x0, -1 < x0 < 1. This generates a logarithmic singularity in the solution u (at the same point x0) and considerably complicates the problem of the numerical approximation of u.
By introducing a nonlinear transformation of the independent variable in the integral equation, we improve the behavior of the unknown function u through incorporation of the Jacobian of the transformation. The price is that the kernel of the new integral equation becomes more complicated. Nevertheless, we are able to show that a Galerkin method based on high-order polynomials is both stable and convergent. We also present some promising numerical results for the corresponding collocation method, for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the Galerkin method.

The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws

Florin Sabac

SIAM J. Numer. Anal. 34, pp. 2306-2318 (13 pages) | Cited 10 times

Online Publication Date: July 25, 2006

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We are interested in the rate of convergence in L1 of the approximate solution of a conservation law generated by a monotone finite difference scheme. Kuznetsov has proved that this rate is 1/2 [USSR Comput. Math. Math. Phys., 16 (1976), pp. 105--119 and Topics Numer. Anal. III, in Proc. Roy. Irish Acad. Conf., Dublin, 1976, pp. 183--197], and recently Teng and Zhang have proved this estimate to be sharp for a linear flux [SIAM J. Numer. Anal., 34 (1997), pp. 959--978]. We prove, by constructing appropriate initial data for the Cauchy problem, that Kuznetsov's estimates are sharp for a nonlinear flux as well.

On the Locking of the Finite Element Method in Thermoelasticity

Eric Boillat

SIAM J. Numer. Anal. 34, pp. 2319-2334 (16 pages)

Online Publication Date: July 25, 2006

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In this article we discuss the solution u to the stationary problem of thermoelasticity and the approximation un of u obtained with a standard Galerkin method. In the incompressible limit, when the Lamé coefficient $\lambda$ becomes infinite, un is subject to a divergence constraint, div un= PnT, where Pn is a given approximation operator acting on the temperature field T. This mechanism provokes a locking phenomenon similar to the one observed in elasticity and studied by Babuska and Suri in Numer. Math., 62 (1992), pp. 439--464.
The purpose of this work is to give a sufficient and necessary condition on the discrete spaces Vn used in the Galerkin formulation, to avoid locking and to get an estimation of the error |u - un|1 which is optimal and independent of $\lambda$.
This criterion will be used to construct two practical examples of methods free from locking for the usual thermoelasticty problem where the temperature field is the solution of a Poisson equation. The first example is based on the h-version of the finite element method; the second one, on the p-version.

Coupling of Mixed Finite Elements and Boundary Elements for A Hyperelastic Interface Problem

Gabriel N. Gatica and Wolfgang L. Wendland

SIAM J. Numer. Anal. 34, pp. 2335-2356 (22 pages) | Cited 14 times

Online Publication Date: July 25, 2006

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We apply the coupling of mixed finite elements and boundary integral methods, using two integral equations on the coupling boundary, to study the weak solvability of a nonlinear elliptic problem arising in elastostatics. Our procedure is based on both the usual Brezzi's theory for linear constrained variational problems and some fundamental tools from nonlinear functional analysis. We derive existence and uniqueness of solution for the continuous variational formulations and provide general approximation results for the corresponding Galerkin schemes. Although we consider bounded domains, the same analysis applies for the case in which the boundary element region is unbounded.

Multidimensional Interpolatory Subdivision Schemes

Sherman D. Riemenschneider and Zuowei Shen

SIAM J. Numer. Anal. 34, pp. 2357-2381 (25 pages) | Cited 20 times

Online Publication Date: July 25, 2006

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This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three-direction box spline Br,r,r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the three-direction box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for r = 2,...,8). Regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness.

A Field-Based Mixed Formulation for the Two-Dimensional Magnetostatic Problem

I. Perugia

SIAM J. Numer. Anal. 34, pp. 2382-2391 (10 pages) | Cited 10 times

Online Publication Date: July 25, 2006

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A mixed variational formulation for the two-dimensional magnetostatic problem involving both the magnetic field and the magnetic displacement field is presented and analyzed. Such a formulation is obtained by minimizing the error in the constitutive equation and imposing as constraints the canonical equations. A numerical discretization by means of "edge" and "face" elements is proposed together with a stability analysis and error estimates.

Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers?

Ivo M. Babuska and Stefan A. Sauter

SIAM J. Numer. Anal. 34, pp. 2392-2423 (32 pages) | Cited 62 times

Online Publication Date: July 25, 2006

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The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM.
However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. Furthermore, we will present a generalized FEM in one dimension which behaves robustly with respect to the wave number.

Global Superconvergence for the Bilinear-Constant Scheme for the Stokes Problem

Jianhua Pan

SIAM J. Numer. Anal. 34, pp. 2424-2430 (7 pages) | Cited 5 times

Online Publication Date: July 25, 2006

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For the stable bilinear-constant scheme applied for the Stokes problem, the accuracy between the interpolation of exact solution and the finite element solution is first examined. Then, the global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method.

Convergence of the Combination Technique for Second-Order Elliptic Differential Equations

Christoph Pflaum

SIAM J. Numer. Anal. 34, pp. 2431-2455 (25 pages) | Cited 2 times

Online Publication Date: July 25, 2006

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The combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization. The advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points.
In this paper, the combination technique is used with a bilinear finite element discretization. Depending on the smoothness of the solution and the coefficients, it is proved for general second-order elliptic differential equations on the unit square that the combined solution converges with order O(h) or O(h log h-1) in the energy norm and with order O(h2 log h-1) or O(h3/2) in the L2-norm, respectively. This holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite. The proof does not use an asymptotic error expansion, but Sobolev space techniques.

On SOR Waveform Relaxation Methods

Jan Janssen and Stefan Vandewalle

SIAM J. Numer. Anal. 34, pp. 2456-2481 (26 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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Waveform relaxation is a numerical method for solving large-scale systems of ordinary differential equations on parallel computers. It differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously. This paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation (SOR) techniques. In particular, different SOR acceleration schemes, based on multiplication with a scalar parameter or convolution with a time-dependent function, are described and theoretically analyzed. The theory is applied to a one-dimensional and two-dimensional model problem and checked against results obtained by numerical experiments.

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

Mario A. Casarin

SIAM J. Numer. Anal. 34, pp. 2482-2502 (21 pages) | Cited 10 times

Online Publication Date: July 25, 2006

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The spectral element method is used to discretize self-adjoint elliptic equations in three-dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with Gauss--Lobatto--Legendre (GLL) quadrature rules of order N, and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q1 or P1 functions, produces a stiffness matrix Kh that is known to be spectrally equivalent to the spectral element stiffness matrix KN. Kh is replaced by a preconditioner $\tilde{K}_h$ which is well adapted to parallel computer architectures. The preconditioned operator is then $\tilde{K}_h^{-1} K_N$.
Techniques for nonregular meshes are developed, which make it possible to estimate the condition number of $\tilde{K}_h^{-1} K_N$, where $\tilde{K}_h$ is a standard finite element preconditioner of Kh , based on the GLL mesh. Two finite element--based preconditioners: the wirebasket method of Smith and the overlapping Schwarz algorithm for the spectral element method are given as examples of the use of these tools. Numerical experiments performed by Pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions.
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