On second-order-accurate discretization of 3D interface problems and its fast solution with a pointwise multigrid solver

摘要

This paper is devoted to developing a complete algorithm for solving a class of 3D elliptic equations with discontinuous coefficients (so-called interface problems). The algorithm is based on a more accurate discretization of the problem, as well as on an efficient solution of the discretized equations. A new seven-point finite volume discretization on cell-centred grids is derived. It is proved that this discretization is second-order accurate in the discrete W_2~1 norm. A multigrid algorithm exploiting pointwise a Jacobi smoother is used to solve the ill-conditioned system of linear algebraic equations arising after the discretization of the above problem. It is demonstrated that the choice of the stopping criterion plays a significant role for the efficiency of the iterative solver. The discretization and the iterative solver are tested in solving an eight-corner problem (i.e. with different diffusivity coefficients in eight subregions). Second-order convergence for both the solution and the flux is observed in numerical experiments. Numerical experiments also demonstrate that the algorithm developed for solving 3D interface problems is robust and fast.