An asymptotic solution approach for elliptic equations with discontinuous coefficients

Abstract

When the coefficients of an elliptic problem have jumps of several orders of magnitude across an embedded interface, many iterative solvers exhibit deteriorated convergence properties or a loss of efficiency and it is difficult to achieve high solution accuracies in the whole domain. In this paper we present an asymptotic solution approach for the elliptic problem ∇⋅(β(x)∇u(x))=f(x) on a domain Ω=Ω+∪Ω− with piecewise constant coefficients β+, β− with β+≫β− and prescribed jump conditions at an embedded interface Γ separating the domains Ω+ and Ω−. We are in particular focusing on a problem related to fluid mechanics, namely incompressible two-phase flow with a large density ratio across the phase boundary, where an accurate solution of the velocity depends on the accurate solution of a pressure Poisson equation with equal local relative errors in the whole domain. Instead of solving the equation in a single solution step we decompose the problem into two consecutive problems based on an asymptotic analysis of the physical problem where each problem is asymptotically independent of the ratio of coefficients ε=β−/β+. The proposed methods lead to a robust and accurate solution of the elliptic problem using standard black-box iterative solvers.