A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients

Abstract

A modified finite difference approximation for interface problems in Rn, n = 1, 2, 3, is presented. The essence of the modification falls in the simultaneous discretization of any two normal components of the flux at the opposite faces of the finite volume. In this way, the continuous normal component of the flux through an interface is approximated by finite differences with second- order consistency. The derived scheme has a minimal (2n + 1)-point stencil for problems in Rn Second-order convergence with respect to the discrete H1-norm is proved for a class of interface problems. Second-order pointwise convergence is observed in a series of numerical experiments with one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) interface problems. The numerical experiments presented demonstrate advantages of the new scheme compared with the known schemes which use arithmetic and harmonic averaging of the discontinuous diffusion coefficient.