任意区域的网格自动生成和任意高阶有限元方法

An arbitrarily high-order nite element method on arbitrarily shaped domains with automatic mesh generation

本文首先简要介绍非拟合网格有限元方法求解复杂区域上椭圆问题的发展现状.然后结合最近本文作者发展的非拟合网格有限元方法,针对二阶椭圆方程提出一种任意光滑区域上的任意高阶协调有限元方法.本文在带悬点的Cartesian网格上自动生成诱导网格,在诱导网格上构造协调的高阶有限元空间,采用Nitsche技术处理Dirichlet边界条件,并证明方法的适定性和hp先验误差估计.数值算例验证了本文的理论结果.

In this paper,we first review the recent progress of unfitted finite element methods for solving elliptic equations on domains with complex geometry.Then we propose an arbitrarily high-order conforming unfitted finite element method defined on automatically generated induced meshes from Cartesian meshes with hanging nodes.The induced mesh allows us to use a simple treatment of the hp finite element method to deal with the hanging nodes to construct a high-order conforming finite element space.Nitsche techniques are used to deal with the Dirichlet boundary conditions.The stability and the hp a priori error estimates of our methods are established.Numerical examples confirm our theoretical findings.