Sixth-Order Hybrid FDMs and/or the M-Matrix Property for Elliptic Interface Problems with Mixed Boundary Conditions
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In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem -∇·( a∇ u)=f in Ω\Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. 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 Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the M-matrix property. Consequently, for the elliptic problem without interface (i.e., Γ 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 Γ, 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 Γ, the two jump functions along Γ, and the functions on ∂Ω. 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 Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l_∞ norm of the proposed hybrid FDMs for the elliptic interface problem.
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