Solving parabolic equations on locally refined grids

An implicit finite difference scheme for solving the heat conduction equation on locally refined grids in a rectangular domain is considered. To solve the resulting system of equations, the conjugate gradient method with preconditioning is used. This method is a variant of the incomplete Cholesky decomposition or modified incomplete Cholesky decomposition. A modification of the computation of the preconditioning matrix for the variant of the incomplete Cholesky-conjugate gradient method for the case of the numerical solution of heat conduction equations with a rapidly varying thermal conductivity coefficient is proposed. Variants of the above-mentioned method designed for use on parallel computer systems with MIMD architecture are proposed. The solution of model problems on a moderate number of processors is used to examine the rate of convergence and the efficiency of the proposed methods.