Elliptic interface problems with discontinuous coefficients appear in many real-world applications: composite materials, fluid mechanics, nuclear waste disposal, and many others. We develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem with discontinuous variable coefficients on a rectangle. The hybrid FDMs utilize a $9$-point compact stencil at any interior regular points and a $13$-point stencil at irregular points near the interface $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order $9$-point compact FDM satisfying the M-matrix property for any mesh size $h>0$. We also derive sixth-order compact FDMs satisfying the M-matrix property for any $h>0$ under various Dirichlet/Neumann/Robin boundary conditions. Thus, considering the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM has the M-matrix property and consequently, satisfies the discrete maximum principle. 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. This is joint work with Bin Han, and Peter Minev.
427 Thackeray Hall