Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

TL;DR

This paper introduces a hybrid finite difference scheme that achieves high-order accuracy for elliptic interface problems with discontinuous, high-contrast coefficients, improving numerical solutions in irregular and regular regions.

Contribution

It develops a hybrid 9-point and 13-point finite difference scheme with sixth and fifth order accuracy for irregular and regular points, respectively, for elliptic interface problems with discontinuous coefficients.

Findings

Numerical experiments confirm sixth order accuracy in $l_2$ and $l_{ abla}$ norms.

The scheme effectively handles high-contrast discontinuous coefficients.

The method demonstrates flexibility across various boundary conditions.

Abstract

For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the interface curve in the porous medium of realistic problems, causing the pollution effect of numerical methods. So, to obtain a reasonable numerical solution of the above problem, the higher order scheme and its effective implementation are necessary. In this paper, we propose an efficient and flexible way to achieve the implementation of a hybrid (9-point scheme with sixth order accuracy for interior regular points and 13-point scheme with fifth order accuracy for interior irregular points) finite difference scheme in uniform meshes for the elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle $\Omega$. We also derive the $6$-point and $4$-point finite difference schemes in uniform meshes with sixth order accuracy for the side points and corner points of various mixed boundary conditions (Dirichlet, Neumann and Robin) of elliptic equations in a rectangle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l_2$ and $l_{\infty}$ norms of the proposed hybrid scheme.