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For elliptic interface problems with discontinuous coefficients, the maximumaccuracy order for compact 9-point finite difference scheme in irregular pointsis three [7]. The discontinuous coefficients usually have abrupt jumps acrossthe interface curve in the porous medium of realistic problems, causing thepollution effect of numerical methods. So, to obtain a reasonable numericalsolution of the above problem, the higher order scheme and its effectiveimplementation are necessary. In this paper, we propose an efficient andflexible way to achieve the implementation of a hybrid (9-point scheme withsixth order accuracy for interior regular points and 13-point scheme with fifthorder accuracy for interior irregular points) finite difference scheme inuniform meshes for the elliptic interface problems with discontinuous andhigh-contrast piecewise smooth coefficients in a rectangle $\Omega$. We alsoderive the $6$-point and $4$-point finite difference schemes in uniform mesheswith sixth order accuracy for the side points and corner points of variousmixed boundary conditions (Dirichlet, Neumann and Robin) of elliptic equationsin a rectangle. Our numerical experiments confirm the flexibility and the sixthorder accuracy in $l_2$ and $l_{\infty}$ norms of the proposed hybrid scheme.

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