满足局部守恒律的Galerkin有限元方法  

A GALERKIN FINITE ELEMENT METHOD SATISFYING LOCAL CONSERVATION LAW

作　　者：何松年[1] 田明[1] 

机构地区：[1]中国民用航空学院理学院,天津300300

出　　处：《石油化工高等学校学报》2005年第4期83-85,共3页Journal of Petrochemical Universities

基　　金：中国民航总局中青年教育研究基金项目资助(01-3-01)

摘　　要：数值求解二阶椭圆边值问题-(β(x)p′(x))′=f(x)往往不仅仅限于计算压力p的逼近解ph,更重要的是寻求流量u=-βp′的逼近解uh。通常采用混合有限元方法得到uh,虽然uh是连续的并且满足局部守恒性质,但产生的系数矩阵是非正定的,通常数值代数方法不易处理,并且计算量大。为此应用满足局部守恒律的Galerkin有限元方法,给出流量u的一种连续分段线性函数形式的有效逼近,一旦p的线性元逼近解ph由标准Galerkin有限元格式求得,则在每一单元上uh可通过基于局部守恒性和流量在整个定义区间上的连续性构造出的极其简单的表达式直接给出。此方法既保持了混合有限元方法的优点,又克服了其缺点。并且分析了这种方法的收敛性和一定条件下的超收敛性,数值计算结果显示了该方法的优越性。Suppose having got the approximate solution to the exact solution of the second order elliptic boundary problem-(β(x)p'(x)}'= f(x) by the standard finite element method, a constructive method for evaluating the approximate flux ph to the exact flux u = -βp' was proposed in this paper. uh is a continuous piecewise linear function, but dealing with nonpositive definits coefficient matrices is very difficult by using usual numerical algebra methods. Thus a constructive method for evaluating the approximate flux ph, Galerkin finite element method satisfying local conservation law is proposed uh. Once ph was got, each finite element uh can be calculated directly via a very simple formula based on the local conservation law and continuity. Thus this method not only maintains local conservation law at the element level, but also saves lots of calculating work. Then the convergence and super convergence of this method were analyzed. Numerical results show its advantages.

关 键 词：有限元方法 局部守恒律 收敛性 超收敛性 

分 类 号：O124[理学—数学] O335[理学—基础数学]