The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let $\Gamma$ be a smooth curve inside a rectangular region $\Omega$. In this paper, we consider the elliptic interface problem $-\nabla\cdot (a \nabla u)=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary conditions, where the coefficient $a$ and the source term $f$ are smooth in $\Omega\setminus \Gamma$ and the two nonzero jump condition functions $[u]$ and $[a\nabla u\cdot \vec{n}]$ across $\Gamma$ are smooth along $\Gamma$. To solve such elliptic interface problems, we propose a high order compact finite difference scheme for numerically computing both the solution $u$ and the gradient $\nabla u$ on uniform Cartesian grids without changing coordinates into local coordinates. Our numerical experiments confirm the fourth order accuracy for computing the solution $u$, the gradient $\nabla u$ and the velocity $a \nabla u$ of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

翻译：与不连续和高多调系数的椭圆界面问题出现在许多应用中,常常导致相应的线性系统出现巨大的条件数量。 因此,非常希望建立高排序计划, 以解决不连续和高多调系数的椭圆界面问题。 让$\ Gamma$在矩形区域中是一个平稳的曲线 $\ Omega$ 和两个非零跳跃条件函数$\ Omega$。 本文认为, 美元之间的椭圆界面问题 $-\nabla\ cddot (a\nablau) = f$, 以美元=Gammaus =f$, 以美元=Gammaus =f$, 以Drichtal $=Gammaus 和源术语 $ 美元=fleglegal 。 我们提议一个高压固定的硬基调的卡通路运算公式, 以美元平价计算我们高基调的平流的平流度和卡路里比的平面的平流计算。