二阶椭圆问题的最小二乘扩展混合有限元方法  

Least-squares Expanded Mixed Finite Element Methods for Second-order Elliptic Problems

作　　者：丁胜[1] 陈焕贞[1] 

机构地区：[1]山东师范大学数学科学学院,济南250014

出　　处：《工程数学学报》2010年第4期669-678,共10页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10271068;10971254);山东省自然科学基金(Y2007A14);山东省优秀中青年科学家科研奖励基金(2008BS01008)~~

摘　　要：为克服最小二乘混合元方法在数值模拟具小扩散系数或低渗透率问题时,应对扩散系数求逆带来的困难,本文基于最小二乘与扩展混合元的思想,对一类刻画扩散、渗透过程的二阶椭圆问题建立了最小二乘扩展混合元格式,证明了格式的稳定性和收敛性质。论证表明,该格式具有无需对小扩散系数求逆,较好的克服了小扩散系数带来的困难;能同时高精度逼近未知函数,梯度及其通量;有限元空间无需满足LBB条件;刚度矩阵对称正定等最小二乘方法和扩展混合元方法的良好性质。数值算例说明了所提算法的有效性。To overcome the diffculties when calculating the inverse of a small diffusive coeffcient when simulating the diffusive problems within a low permeability zone by a least-squares mixed finite element method, we use an expanded mixed finite element instead of a mixed finite element and de- velop a least-squares expanded mixed finite method for second-order elliptic problems. We prove the stability and convergence for the proposed procedure. The analysis shows that the method inherits the advantages of least-squares and expanded mixed finite element methods, such as it is not necessary to calculate the inverse of a small diffusive coeffcient; approximating the unknown, its gradient and its flux directly; the finite element space being free of LBB condition as well as the stiff matrix being symmetry and positive-definite. Numerical tests are performed to confirm the theoretical analysis.

关 键 词：二阶椭圆问题 最小二乘方法 扩展混合有限元方法 最优误差估计 

分 类 号：O241.82[理学—计算数学]