Composite iterative method for elliptic problems in irregular regions

Abstract

A finite element piecewise linear approximation to a self-adjoint second-order elliptic equation in an irregular bounded two-dimensional region with Dirichlet boundary condition is considered. A composite inner-outer iterations method for the arising linear systems of algebraic equations is presented. Preconditioned Richardson iterations are chosen as the outer iterations, in which the discrete Laplacian is employed as a preconditioner. In each outer iteration, systems with the Laplacian in an irregular region are solved by the imbedding capacitance matrix method that uses conjugate gradients as the inner iterations. We prove that with an optimal control of the termination criterium for inner iterations the rate of convergence of this composite iterative process depends only logarithmically on h, the parameter of triangulation. Extensive numerical experiments illustrate the theoretical predictions.