We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constraints for DG method on how the interface intersects to the elements in the mesh. Other than enjoying the advantages of DG method, our method may achieve even better efficiency than the conforming finite element method, as illustrated by numerical examples.

翻译：我们建议采用不连续的Galerkin(DG)方法,使用新的近似空间来估计不合适的网格的椭圆界面问题。近似空间是通过对每个元素自由度的补丁重建构建的。在L2规范中和DG能源规范中的最佳误差估计都是在没有DG方法在界面如何与网格中元素交叉方面的典型限制的情况下取得的。除了享受DG方法的好处外,我们的方法可能比符合的有限元素方法效率更高,如数字示例所示。