Composite iterative method for elliptic problems in irregular regions
A finite element piecewise linear approximation with triangular elements 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 system 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. The inner iterations are terminated before they have converged to full accuracy. The authors prove that with an optimal control of the termination criterion for inner iterations the rate of convergence of this composite iterative process depends only logarithmically on h, the parameter of triangulation. Numerical examples are shown.
self-adjoint second-order elliptic equation
scientific article; zbMATH DE number 4151773
scientific article