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