弱有限元方法简论

BASICS OF WEAK GALERKIN FINITE ELEMENT METHODS

本文简述弱有限元方法(weak Galerkin finite element met,hods)的数学基本原理和计算机实现.弱有限元方法对间断函数引入广义弱微分,并将其应用于偏微分方程相应的变分形式进行数值求解,而数值解的弱连续性则通过稳定子或光滑子来实现.弱有限元方法针对广义函数而构建,是经典有限元方法的一种自然拓广,且能够弥补经典有限元方法的某些缺憾,也因此在科学与工程计算领域具有广泛的应用前景.

This article introduces the basic principles and some recent developments of weak Galerkin finite element methods （WG-FEM）, including their mathematical theory and com- puter implementation. The WG-FEM, by design, makes use of discontinuous piecewise polynomials on finite element partitions consisting of polygons and polyhedrons of arbitrary shape. The weak Galerkin methods provide necessary weak continuities for the approximate solutions by using weakly defined derivatives and parameter-free stabilizers. The WG-FEM is a natural extension of the standard Galerkin finite element method （FEM）, and is advan- tageous over the standard FEM in some applications. The weak Galerkin methods thus have great potentials in applied problems arising from science and engineering.