On the Additive Version of the Algebraic Multilevel Iteration Method for Anisotropic Elliptic Problems

Abstract

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.

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

“…In the latter case, a stabilization is done on some of the intermediate levels by solving the corresponding system with the same preconditioned method to a lower accuracy 10 −3 , which requires 3-4 inner iterations. An analogous way of stabilizing an AMLI-type solver of additive form is used in [10].…”

A component decomposition preconditioning for 3D stress analysis problems

“…are anisotropic (in the second and third term in (33) we have the ÿrst derivatives only in one direction). In Reference [8] Axelsson discusses the implications of this fact on the e ciency and robustness of the AMLI methods (References [34,35]) when applied as the local subproblem solvers.…”

Two Simple Derivations of Universal Bounds for the C.B.S. Inequality Constant

“…However, for nearly degenerate triangles or, equivalently, strongly anisotropic coefficients it becomes very ill-conditioned and requires some special element by element preconditioner, see e.g. [7] for details.…”