论文标题
局部各向异性拟合混合网的有限元方法
Finite Element Methods For Interface Problems On Local Anisotropic Fitting Mixed Meshes
论文作者
论文摘要
本文开发了一种简单有效的接口构成的网格生成算法。该算法可以产生局部各向异性拟合混合网，该网格由界面附近的三角形和四边形组成。提出了一种新的有限元方法，用于基于结果网格的二阶椭圆界面问题。各向异性元素上的最佳近似功能在$ h^1 $和$ l^2 $规范中都证明了。离散系统通常由于各向异性和界面附近的小元素而受到不良条件。于是，提出了一种多方方法来解决此问题。相对于系数跳跃比和网格尺寸，多机方法的收敛速率被证明是最佳的。提出了数值实验以证明理论结果。
A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new finite element method is proposed for second order elliptic interface problems based on the resulting mesh. Optimal approximation capabilities on anisotropic elements are proved in both the $H^1$ and $L^2$ norms. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a 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.
