椭圆型偏微分方程的弱有限元方法 献给林群教授80华诞

Weak Galerkin finite element methods for elliptic PDEs

弱有限元方法 (weak Galerkin finite element methods),简称WG方法,是求解偏微分方程的一种全新且高效的数值方法.WG有限元方法的主要思想是,对间断函数引入广义弱微分算子,并将其应用于通常的变分形式中以对相应的偏微分方程进行数值求解,数值解的连续性则通过稳定子以弱形式来实现.本文以二阶椭圆问题为例,详细介绍弱有限元方法的原理和基础,并给出相关的理论分析.此外,本文简单介绍其他椭圆问题的弱有限元方法的离散格式.弱有限元方法的最大特点是,(1)有限元剖分允许任意多边形或多面体;(2)总体刚度矩阵可通过单元刚度矩阵叠加而得;(3)逼近函数构造简单,且极易满足相应的稳定性条件;(4)格式可做杂交处理以并行消去单元内部自由度.

The weak Galerkin finite element method （WG） is a newly developed and efficient numerical tech- nique for solving partial differential equations （PDEs）. The central idea of WG is to interprete partial differential operators as generalized distributions, called weak differential operators, over the space of discontinuous functions including boundary information. The weak differential operators are further discretized and applied to the cor- responding variational formulations of the underlying PDEs. This paper introduces the basic principle and the theoretical foundation for the WG method by using the second order elliptic equation. The WG method is further applied to several other model equations, such as the Stokes, biharmonic, and Maxwell equations to demonstrate its power and efficiency as an emerging new numerical method. The main strengths of the WG method are： （1） the finite element partition can be of polytopal type; （2） the global stiffness matrix can be assembled by adding up local stiffness matrices; （3） the weak finite element space is easy to construct with any given stability and approximation requirement; and （4） the WG schemes can be hybridized so that some unknowns associated with the interior of each element can be locally eliminated, yielding a system of linear equations involving much less number of unknowns than what it appears.