Sixth-Order Hybrid FDMs and/or the M-Matrix Property for Elliptic Interface Problems with Mixed Boundary Conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem $-\nabla \cdot( a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface inside $\Omega$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $\Gamma$. The hybrid FDMs utilize a 9-point compact stencil at any interior regular point of the grid and a 13-point stencil at irregular points near $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order 9-point compact FDM satisfying the M-matrix property. Consequently, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs having the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. For irregular points near $\Gamma$, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient $a$, the source term $f$, the interface curve $\Gamma$, the two jump functions along $\Gamma$, and the functions on $\partial \Omega$. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing $\Gamma$ and are also applicable if the jump conditions on $\Gamma$ only depend on the geometry (e.g., curvature) of the curve $\Gamma$. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.