A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let Γ be a smooth curve inside a rectangular region Ω. In this paper, we consider the elliptic interface problem −∇·(a∇u)=f in Ω∖Γ with Dirichlet boundary conditions, where the coefficient a and the source term f are smooth in Ω∖Γ and the two jump condition functions [u] and [a∇u·n→] across Γ are smooth along the interface Γ. To solve such elliptic interface problems, we propose a high order compact 9-point finite difference scheme and a high order local calculation for numerically computing the solution u and its gradient ∇u respectively on uniform Cartesian grids without changing coordinates into local coordinates. We numerically verify the sign conditions of our proposed compact finite difference scheme and prove the convergence rate by the discrete maximum principle. Our numerical experiments confirm the fourth order accuracy for computing the solution u in both l₂ and l_∞ norms of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.