A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces

Abstract

We present a finite volume method for the solution of the two-dimensional elliptic equation @?.(@b(x)@?u(x))=f(x) with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution u(x) resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the L^~ and L^2 norm.