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

Highlights：

• We prove that the maximum order of the compact 9-point finite difference schemes for the numerical approximated solutions at irregular points on uniform meshes is three.

• We numerically verify the sign conditions of our proposed compact finite difference scheme and prove the convergence rate by the discrete maximum principle.

• We compare our proposed compact scheme with the known schemes IIM, EJIIM, MIB and AMIB to present the efficiency and advantages of our proposed scheme.

• Our numerical experiments confirm the fourth order accuracy for computing the solutions u in both l2 and l∞ norms of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous, variable, and high-contrast coefficients.

摘要

•We construct a high order compact 9-point finite difference scheme for the numerical solutions on uniform meshes for elliptic interface problems with discontinuous, piecewise smooth and high-contrast coefficients, discontinuous source terms and two non-homogeneous jump conditions.•We prove that the maximum order of the compact 9-point finite difference schemes for the numerical approximated solutions at irregular points on uniform meshes is three.•We numerically verify the sign conditions of our proposed compact finite difference scheme and prove the convergence rate by the discrete maximum principle.•We compare our proposed compact scheme with the known schemes IIM, EJIIM, MIB and AMIB to present the efficiency and advantages of our proposed scheme.•Our numerical experiments confirm the fourth order accuracy for computing the solutions u in both l2 and l∞ norms of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous, variable, and high-contrast coefficients.