Immersed Interface Method for elliptic equations based on a piecewise second order polynomial

Abstract

In this paper, we develop finite difference methods for elliptic equations in a domain Ω∈Rd, d=1,2. Within the region Ω, we suppose there is an irregular surface Γ of codimension 1 (hereafter called an interface) across which the function u or some of its derivatives are known to be discontinuous. We use uniform grid and a piecewise second order polynomial to approximate u, then we get a second order method for these problems. At last, we give several examples to show the correctness and efficiency of the scheme.