Konvergenzaussagen für Kollokationsverfahren bei elliptischen Randwertaufgaben

Summary We examine the convergence of collocation with polynomials for elliptic problems. The general theory is imbedded in theL ^p -theory from which known results of existence and regularity are adopted. Convergence results are deduced in the cases of periodic, Dirichlet and mixed boundary conditions. It becomes obvious that the order of convergence essentially depends on the smoothness of the solution.

Dieser Aufsatz enthält Ergebnisse aufbauend auf der Dissertation des Verfassers, die von Prof. Dr. K. Witsch, Universität Düsseldorf, angeregt und unterstützt wurde     

On constrained Newton linearization and multigrid for variational inequalities

Summary. We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments. Received March 22, 1999 / Revised version received February 24, 2001 / Published online October 17, 2001     

Convergence analysis of a multigrid method for convection–diffusion equations

Summary. This paper is concerned with the convergence analysis of robust multigrid methods for convection-diffusion problems. We consider a finite difference discretization of a 2D model convection-diffusion problem with constant coefficients and Dirichlet boundary conditions. For the approximate solution of this discrete problem a multigrid method based on semicoarsening, matrix-dependent prolongation and restriction and line smoothers is applied. For a multigrid W-cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion/convection ratio. For the contraction number of a multigrid V-cycle a bound is proved which is uniform for a class of convection-dominated problems. The analysis is based on linear algebra arguments only. Received April 26, 2000 / Published online June 20, 2001     

Anmerkungen zu einem Mehrgitterverfahren für lineare Komplementaritätsprobleme

Zusammenfassung Es wird ein Mehrgitterverfahren zur Lösung großer, dünn besetzter, symmetrischer, positiv definierter, linearer Komplimentaritätsprobleme beschrieben. Die Konvergenz des Verfahrens kann bewiesen werden.

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Notes on a multigrid method for linear complementarity problems

Summary A multigrid method for the solution of large, sparse, symmetric, positive definite linear complementarity problems is described. The convergence of the method can be proved.

     

The Convergence of the Multigrid Algorithm for Navier-Stokes Equations

This paper deal with a multigrid algorithm for the numerical solution of Navier-Stokes problems. The convergence proof and estimation of the contraction number of the multigrid algorithm are given.     

A multigrid method for a Petrov-Galerkin discretization of the Stokes equations

Summary We introduce a multigrid method for the solution of the discrete Stokes equations, arising from a Petrov-Galerkin formulation. The stiffness matrix is nonsymmetric but coercive, hence we consider smoothing iterations which are not suitable for usual indefinite problems. In this report, we prove convergence for a multigrid method with Richardson iteration in the smoothing part.     

On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse‐grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse‐grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W‐cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000     

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number. Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000     

Multigrid for the Galerkin least squares method in linear elasticity

In SIAM J. Numer. Anal. 28 (1991) 1680-1697, Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid method. This multigrid is robust in that the convergence is uniform as the parameter ν goes to 1/2. Computational experiments are included.     

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

Sparse-grid finite-volume multigrid for 3D-problems

We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset of all possible meshes in the hierarchy, a sparse grid finite-volume discretisation can be constructed.The multigrid algorithm consists of a simple damped point-Jacobi relaxation as the smoothing procedure, while the coarse grid correction is made by interpolation from several coarser grid levels.The combination of sparse grids and multigrid with semi-coarsening leads to a relatively small number of degrees of freedom,N, to obtain an accurate approximation, together with anO(N) method for the solution. The algorithm is symmetric with respect to the three coordinate directions and it is fit for combination with adaptive techniques.To analyse the convergence of the multigrid algorithm we develop the necessary Fourier analysis tools. All techniques, designed for 3D-problems, can also be applied for the 2D case, and — for simplicity — we apply the tools to study the convergence behaviour for the anisotropic Poisson equation for this 2D case.     

Consistent approximations in Newton-type decomposition methods

Summary A class of Newton-type decomposition methods for the solution of large systems of nonlinear equations recently introduced by the first two authors is extended in such a way that consistent approximations to the derivatives by appropriate difference quotients are permitted. Concrete realizations with essential merits for practical computation and a rigorous convergence analysis are given. For special choices of the discretization parameterR-orders greater than one are derived.Dedicated to Prof. Dr. Dr. h.c. Lothar Collatz on the occasion of his 75th birthday     

Multigrid methods with web-splines

Summary. We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weightedextended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width. Received August 17, 2000 / Published online August 17, 2001     

Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

论非线性多重网格法的逼近性质

多重网格法是一种求解椭圆边值问题离散所得的大型线性或非线性方程组的“最优”解法。在有限元离散情形,Hackbusch提出了一种多重网格法的收敛分析方法,即把线性或非线性的多重网格法收敛率的估计问题归结为所谓“光滑性质”与“逼近性质”的研究。在线性情形,若已知有限元解的误差估计,一般容易得到多重网格法的“逼近性质”。但对非线性多重网格法的“逼近性质”在什么条件下成立,尚未见到这方面的工     

Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities

Summary. Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset. One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions. Received December 13, 2001 / Revised version received February 19, 2002 / Published online June 17, 2002 This work was partially supported by the Norwegian Research Council under projects 128224/431 and SEP-115837/431.     

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

We present a new strategy to accelerate the convergence rate of a high‐accuracy multigrid method for the numerical solution of the convection‐diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss‐Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourth‐order compact discretization are investigated numerically. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 1–10, 2000     

Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations

The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations.     

Convergence analysis of multigrid methods with residual scaling techniques

In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper.     

On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity

Hemivariational inequalities have been successfully employed for mathematical and numerical studies of application problems involving nonsmooth, nonmonotone and multivalued relations. In recent years, error estimates have been derived for numerical solutions of hemivariational inequalities under additional solution regularity assumptions. Since the solution regularity properties have not been rigorously proved for hemivariational inequalities, it is important to explore the convergence of numerical solutions of hemivariational inequalities without assuming additional solution regularity. In this paper, we present a general convergence result enhancing existing results in the literature.