【作者】 李清善；

【导师】 陈绍春； 石东洋；

【作者基本信息】 郑州大学 ， 基础数学， 2000， 硕士

【摘要】 正则性条件是传统有限元的本质性条件，这使得有限元的应用受到了很大的限制。本文在非正则剖分条件下，研究了任意窄四边形上的类Wilson元。主要结果有： 1．通过参考元上Wilson元的构造，证明了由此产生的有限元对任意窄四边形网格剖分通过Irons分片检查。 2．利用窄四边形等参有限元的插值定理，得到了窄四边形上类Wilson元的插值误差。 3．通过参考元K＝[0，1]×[0，1]上的一系列估计，证明了在不满足正则性假定条件下二阶问题的收敛性，而且其收敛阶与正则剖分条件下相同。同时，通过对参考元上Poincare不等式的精细证明，本文具体给出了各估计式中的常数。更多还原

【Abstract】 The regularity conditions are essential for the tranditional finite elements. They restrict the application of the elements in a way. In the case of non-regularity meshes, the quasi-Wilson element for arbitrary narrow quadrilateral are studied in this paper. The main results are as follows: 1. By construction of quasi-Wilson element on reference element, we prove that the resulting elements pass through Irons? patch test for arbitrary narrow quadrilateral meshes. 2. Using the interpolation theorems for narrow quadrilateral isoparametric finite elements, the interpolation errors of the quasi- Wilson element for arbitrary narrow quadrilateral are obtained. 3. By a series of estimates on reference element, the convergence property for second order problems are proved without satisfying the regularity conditions, and its convergence order is same as that of regularity meshes. Meanwhile, by the accurate proof of Poincare inequality on reference element, some constants of estimates are given concretely.更多还原