A circulant preconditioner for domain decomposition algorithm for the solution of the elliptic problems

Abstract

A preconditioned conjugate gradient (PCG)-based domain decomposition method was given in [11] and [1.2] for the solution of linear equations arising in the finite element method applied to the elliptic Neumann problem. The novelty of the proposed algorithm was that the recommended preconditioner was constructed by using symmetric-cyclic matrix. But we could give only the definitions of the entries of this cyclic matrix. Here we give a short description of this algorithm, the method of calculation of matrix entries and the results of calculation. The numerical experiments presented show, that this construction of precondition in the practice works well.

An H‐matrix Type Preconditioner For Frictional Contact Problems

“…The central aspect of this work is the adaptation of a preconditioner construction developed by B. Kiss at al., in [2,3,5] for non-overlapping Dirichlet type domain decomposition method to the contact problem. Our preconditioner construction is based on the application of the H-matrix technique [6,7] together with the representation of the H 1/2 seminorm by a sum of partial seminorms [4].…”

“…In Section 3, for a given Γ c we apply the DD-method which consists of computing the solution as follows [2,3]:…”

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The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non-overlapping domain decomposition method [2,3]. The global problem is transformed to a independant local problems posed in each bodie and a problem posed on the contact surface (the interface problem).

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DD-type Domain Decomposition Method for Frictional Contact Problem

“…Kiss at al., in [2] [3] for non-overlapping Dirichlet type domain decomposition method to the contact problem. The mechanical interaction between the bodies is modeled, under the assumption of small displacement, by the bilateral or unilateral contact condition, and Coulomb´s friction law relating the contact force and the displacement.…”

Domain-Decomposition based H1/2 Seminorm Preconditioners for Frictional Contact Problems

“…An algorithm is introduced to solve the resulting finite element system by a non-overlapping domain decomposition method, which consists of a suitable iterations based on the solutions of the elasticity equations for each body separately and the solution of a smaller problem on the contact surface. The central aspect of this work is the adaptation of a preconditioner construction developed in [2,3,4] for non-overlapping Dirichlet type domain decomposition method to the contact problem. The circulant matrix representations of the H 1 2 seminorm has been proved to be spectrally equivalent to the Schur Complement in [8].…”