Inexact Solvers Research Articles

ON THE OPTIMALITY OF SOME FAC AND AFAC METHODS FOR ELLIPTIC FINITE ELEMENT PROBLEMS

We consider some solution methods for large sparse linear systems of equations which arise from second-order elliptic finite element problems defined on composite meshes. Historically these methods were called FAC and AFAC methods. Optimal bounds of the condition number for certain AFAC iterative operator are established by proving a strengthened Cauchy-Schwarz inequality using an interpolation theorem for Hilbert scales. This work completes earlier work by Dryja and Widlund. We also apply an extension theorem for finite element functions to get a weaker bound under some more general assumptions. The optimality of the FAC methods, with exact solvers or spectrally equivalent inexact solvers being used, is also proved by using similar techniques and some ideas from multigrid theory..

ON THE OPTIMALITY OF SOME FAC AND AFAC METHODS FOR ELLIPTIC FINITE ELEMENT PROBLEMS

We consider some solution methods for large sparse linear systems of equations which arise from second-order elliptic finite element problems defined on composite meshes. Historically these methods were called FAC and AFAC methods. Optimal bounds of the condition number for certain AFAC iterative operator are established by proving a strengthened Cauchy-Schwarz inequality using an interpolation theorem for Hilbert scales. This work completes earlier work by Dryja and Widlund. We also apply an extension theorem for finite element functions to get a weaker bound under some more general assumptions. The optimality of the FAC methods, with exact solvers or spectrally equivalent inexact solvers being used, is also proved by using similar techniques and some ideas from multigrid theory..

Inexact solvers on element interfaces for the p and h- p finite element method

Parallel and iterative solvers for large-scale linear system arising from finite element discretization, based on domain decomposition, decompose the approximation spaces into a number of subspaces. The corresponding subproblems are solved in these subspaces, and the solutions are corrected in an iterative process. The subproblems are much smaller in size and much easier to be solved. One of these subspaces is spanned by (high order) element side modes in two dimensions and (high order) element face modes in three dimensions. Solving the subproblems on the element interfaces is a major cost in each step of iteration. To reduce computational cost, various inexact solvers on element interfaces are proposed in this paper, with theoretical analysis and numerical illustrations.

Inexact Newton solvers in plasticity: theory and experiments

Applications of inexact Newton and inexact Newton-like solvers are described and analysed for the solution of non-linear systems arising in the numerical solution of problems of elastoplasticity. Both explicit and return mapping incremental finite element algorithms are considered. © 1997 John Wiley & Sons, Ltd.

On a relaxation procedure for domain decomposition methods with many subdomains and inexact solver

A domain decomposition technique was presented by Marini and Quarteroni [9,10]. They assumed that the domain was partioned into two nonintersecting subdomains. In this paper we generalized this techniques such that the domain can be partioned into many subdomains. Furthermore, we discuss the domain decomposition technique with domain partioned into two subdomains when the solution on one subdomain is inexact.

The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains

We consider the Neumann-Dirichlet domain decomposition method for the solution of linear elliptic boundary value problems. We study the following question. Suppose that the auxiliary problems on the subdomains are not solved exactly, but only with a fixed, mesh size independent accuracy. Does the speed of convergence remain mesh size independently bounded? We show that the answer is no in general, but that mesh size independent convergence can be obtained if the accuracy requirement on the subsolvers becomes increasingly severe as the mesh size tends to zero.