椭圆型方程的一种新型混合有限元格式  

A New Mixed Finite Element Scheme for Elliptic Equations

作　　者：史峰[1] 于佳平[1] 李开泰[1] 

机构地区：[1]西安交通大学理学院,西安710049

出　　处：《工程数学学报》2011年第2期231-237,共7页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10971165;10771167;10926080)~~

摘　　要：基于实际问题对通量较低的正则性要求,本文建立了椭圆型方程的一种新型混合变分形式.在鞍点问题框架下,通过验证LBB条件,证明了该混合形式解的存在唯一性.由于压力空间不再是传统的H(div),而是平方可积空间,因此混合元的选取变得简单容易.同时本文给出了相应的有限元逼近形式,并对于由分片常数速度元和分片线性压力元构成的协调有限元对P20P1,通过验证速度投影算子的有界性,证明了有限元解的存在唯一性,以及有限元逼近在某种意义下是最优的.最后给出了数值算例,验证了方法的有效性和理论分析的正确性.此方法还可以通过增加简单的稳定项使用最为常用的最低阶等阶有限元.In this paper,a new mixed variational formulation to the elliptic equation is given based on the less regularity of flux in practical problems,and the existence and uniqueness of solution to this formulation is shown under the framework of the saddle point problem by verifying the LBB condition.Since in the new mixed variational formulation,the pressure belongs to the square integrable space instead of the classical H （div）,the choices of finite element pairs become simple and easy.Based on this new formulation,the corresponding conforming finite element approximation is addressed,and the existence and uniqueness of finite element solutions are obtained for P 2 0 P 1 pair consisting of piecewise constant element for velocity and piecewise linear element for pressure,by proving the boundedness of projection operator for velocity.Moreover,it＇s proven that the finite element approximation is optimal in some sense.Finally,some numerical experiments verify the effectiveness and theoretical results of our method.This method can be extended to utilize the lowest equal order finite element pair by adding some simple stabilization terms.

关 键 词：椭圆型方程 混合变分形式 协调有限元对 LBB条件 

分 类 号：O175.25[理学—数学] O241.82[理学—基础数学]