2023/06/22 by Qiwei Feng, Bin Han, Feng, Qiwei +3 · 2 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2306.13001
openalex publication_date 2023/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem -∇ ⋅( a∇ u)=f in Ω\backslash 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 points 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 sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. 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 numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.