解重调和问题混合有限元方程的直接方法

§1.引言 考虑如下重调和方程的齐次边值问题: △~2w=f,在Ω中, w=?w/?v=0,在Ω上.(1.1)其中Ω是平面凸多边形区域,?Ω是Ω的边界,?/?v表示?Ω上的外法向导数.

A DIRECT METHOD FOR SOLVING THE DISCRETE MIXED FINITE ELEMENT EQUATIONS OF THE BIHARMONIC PROBLEM

In this paper, a direct method for solving the discrete mixed finite element equa-tions of the biharmonic problem with homogeneous boundary values is given, based onthe abstract analysis for the mixed finite element method (c.f. 1], 2] and 6] ). Let?_h be a regular triangulation of a convex polygon in plane, N and M be the numbersof the inter-nodes and boundary nodes respectively. Then the procedure for solving thediscrete mixed finite element equations of the biharmonic problem can be deduced tosolving three systems of dimension N and two systems of dimension M essentially, andthe coefficient matrix of these systems is symmetric positive definite. The conditionnumbers of the systems are given.