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# Solving elliptic problems by the domain decomposition method using preconditioning matrices derived by multilevel splittings of the finite element matrix

## Abstract

We study some aspects of domain decomposition technique applied to a multilevel iterative method in solving second order elliptic problems by the finite element method. This iterative method makes use of a preconditioning matrix which is constructed by using a sequence of nested discretizations of the considered elliptic problem. The main computational task in solving problems with this preconditioning matrix is solution of problems, on each fixed level, with matrices which have condition number bounded on h_{k} (— the discretization parameter on the level k). Here we propose to use, in connection with parallel implementation of this multilevel iterative method, domain decomposition technique in order to solve these problems with constantly conditioned matrices. Numerical tests are presented and they are compared with tests in which the solution method is the preconditioned conjugate gradient method with preconditioning matrices derived by approximate (blockwise) factorizations of the constantly conditioned matrices. The last methods suffers of the fact that it is not well parallelizable but it is very fast convergent.

## Keywords

Elliptic Problem Domain Decomposition Conjugate Gradient Method Multigrid Method Domain Decomposition Method## Preview

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