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The elliptic interface problems with discontinuous and high-contrastcoefficients appear in many applications and often lead to huge conditionnumbers of the corresponding linear systems. Thus, it is highly desired toconstruct high order schemes to solve the elliptic interface problems withdiscontinuous and high-contrast coefficients. Let $\Gamma$ be a smooth curveinside a rectangular region $\Omega$. In this paper, we consider the ellipticinterface problem $-\nabla\cdot (a \nabla u)=f$ in $\Omega\setminus \Gamma$with Dirichlet boundary conditions, where the coefficient $a$ and the sourceterm $f$ are smooth in $\Omega\setminus \Gamma$ and the two nonzero jumpcondition functions $[u]$ and $[a\nabla u\cdot \vec{n}]$ across $\Gamma$ aresmooth along $\Gamma$. To solve such elliptic interface problems, we propose ahigh order compact finite difference scheme for numerically computing both thesolution $u$ and the gradient $\nabla u$ on uniform Cartesian grids withoutchanging coordinates into local coordinates. Our numerical experiments confirmthe fourth order accuracy for computing the solution $u$, the gradient $\nablau$ and the velocity $a \nabla u$ of the proposed compact finite differencescheme on uniform meshes for the elliptic interface problems with discontinuousand high-contrast coefficients.

A High Order Compact Finite Difference Scheme for Elliptic Interface Problems with Discontinuous and High-Contrast Coefficients