A new finite element method is proposed for second order elliptic interface problems based on a local anisotropic fitting mixed mesh. The local anisotropic fitting mixed mesh is generated from an interface-unfitted mesh by simply connecting the intersected points of the interface and the underlying mesh successively. Optimal approximation capabilities on anisotropic elements are proved, the convergence rates are linear and quadratic in \(H^1\) and \(L^2\) norms, respectively. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a new multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results.

提出了一种基于局部各向异性拟合混合网格的二阶椭圆界面问题有限元方法。局部各向异性拟合混合网格是由界面未拟合的网格通过简单地依次连接界面和底层网格的相交点来生成的。证明了对各向异性单元的最优逼近能力，收敛速度在\(H^1\)和\(L^2\)中是线性和二次的规范，分别。由于界面附近的各向异性和小元素，离散系统通常是病态的。于是，提出了一种新的多重网格方法来处理这个问题。多重网格方法的收敛速度在系数跳跃比和网格尺寸方面均显示为最佳。给出了数值实验来证明理论结果。