In this paper a recently proposed additive version of the algebraic multilevel iteration method for iterative solution of elliptic boundary value problems is studied. The method constructs a nearly optimal order parameter-free preconditioner, which is robust with respect to anisotropy and discontinuity of the problem coefficients. It uses a new strategy for approximating the blocks corresponding to "new" basis functions on each discretization level. To cope with the difficulties arising from the anisotropy, the problem on the coarsest mesh is solved using a bordering technique with a special choice of bordering vectors. The aim is to find a parameter-free "black-box" robust solver.The results are derived in the framework of a hierarchical basis, linear finite element discretization of an elliptic problem on arbitrary triangular meshes, and a hierarchical basis, bilinear finite element discretization on Cartesian meshes.A comparison of the method with some other iterative solution techniques is presented. Robustness and high efficiency of the proposed algorithm are demonstrated on several model-type problems.Key words. multilevel iterative methods, optimal order preconditioning, anisotropic elliptic problems, black-box solvers AMS subject classifications. 65F10, 65F15, 65N55 PII. S1064827597320058 Introduction.There is a strong demand for efficient iterative methods for the numerical solution of second order elliptic boundary value problems arising in material science, computational fluid dynamics, medicine, and many other branches of scientific research and engineering. Handling problems with irregularity of the discretization meshes, anisotropy, and discontinuity of the coefficient function in the problem are challenges for iterative solution methods. Many efforts have been devoted to the construction of efficient and robust techniques able to overcome some of those difficulties (see [4], [17], [19], [22], for instance), but there is still no common solution to all arising questions. The framework of algebraic multilevel iteration (AMLI) methods (see [1], [2], [14], [15], for instance) has proved to be very suitable for development of the solvers which are robust in the above mentioned sense. An important feature of these methods is that their convergence rate is independent of the regularity of the elliptic problem. It has been proven that the AMLI methods are optimal (or nearly optimal) with respect to both convergence rate and arithmetic cost per iteration for problems with discontinuous coefficients and with respect to size and direction of anisotropy [4], [22]. The properties of the AMLI methods depend mainly on a single parameter γ, which is the constant in the so-called strengthened Cauchy-Bunyakowski-Schwarz inequality; the parameter γ can be computed at the local element level and, therefore, does not depend on the jumps of the coefficients between elements.
A le x a n d e r Padiy* O w e Axelsson* B en Polm an* A b s tra c tThe present work is devoted to a class of preconditioners based on the augmented matrix approach considered earlier by two of the present authors. It presents some gener alizations of the subspace-correction schemes studied earlier and gives a brief comparison of the developed technique with a somewhat similar "deflation" algorithm.The developed preconditioners are able to improve significantly an eigenvalue dis tribution of certain severely ill-conditioned algebraic systems by using properly chosen projection matrices, which correct the low-frequency components in the spectrum. One of the main advantages of the proposed approach is the possibility to use inexact solvers within the projectors. Another attractive feature of the developed method is that it can be easily combined with other preconditioners, for instance those which correct the high-frequency eigenmodes.
The present work is devoted to the damped Newton method applied for solving a class of non‐linear elasticity problems. Following the approach suggested in earlier related publications, we consider a two‐level procedure which involves (i) solving the non‐linear problem on a coarse mesh, (ii) interpolating the coarse‐mesh solution to the fine mesh, (iii) performing non‐linear iterations on the fine mesh. Numerical experiments suggest that in the case when one is interested in the minimization of the L2‐norm of the error rather than in the minimization of the residual norm the coarse‐mesh solution gives sufficiently accurate approximation to the displacement field on the fine mesh, and only a few (or even just one) of the costly non‐linear iterations on the fine mesh are needed to achieve an acceptable accuracy of the solution (the accuracy which is of the same order as the accuracy of the Galerkin solution on the fine mesh). Copyright © 2000 John Wiley & Sons, Ltd.
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