四阶椭圆方程的三棱柱元分析

Triangular Prism Element Analysis of Fourth Order Elliptic Equations

关于四阶椭圆方程构造合适的有限元空间,该问题在二维空间中得到了较广泛的研究,但在三维空间中取得的成果还不是很多.四阶问题三棱柱单元的构造不仅在数学理论上重要,其重要性在应用领域也有所体现.本文构造出了一个23-参数非协调三棱柱单元,并证明了该单元关于三维四阶椭圆方程收敛.为保证单元的适定性,形函数空间的选取借助了泡函数.

The construction of appropriate finite element spaces for fourth order elliptic partial differential equations is an intriguing subject. This problem has been well studied in two-dimensional spaces. In comparison, there has been very little work devoted to three-dimensional problems. The construction of triangular prism finite element for fourth order problem is not only important from a mathematical point of view but also in practical applications. In this paper, a 23-parameter triangular prism nonconforming finite element are proposed and proved to be convergent for a model biharmonic equation in three dimensions. In order to ensure the well posedness of the element, the shape function space is selected by using the bubble functions .