Multigrid Methods for Nonconforming Finite Elements

有限元多重网格法是求解椭圆型边值问题的最优阶算法。近似解的误差在全局误差的理论范围内，但计算成本仅与离散方程中的未知量成正比。直到最近，这一领域的大部分研究主要集中在嵌套一致性有限元空间上。然而，非一致性有限元在应用中经常使用，因为它们更容易编程。在之前的工作中，主要研究者在多边形域上对P1非协调有限元应用于泊松方程和Morley有限元应用于双调和方程进行了研究，得到了非协调有限元的全w周多网格收敛的结果。在本项目中，这些结果将应用于平稳Stokes方程，并扩展到光滑边界域。多网格算法将用于线弹性和板弯曲的非一致性方法，以及等效于非一致性方法的混合方法。研究了v循环非协调方法的收敛性。我们将对每一个问题进行数值实验。

8904911 Brenner The multigrid method for finite elements is an optimal order algorithm for solving elliptic boundary value problems. The error of the approximate solution lies within the theoretical bound for the global error, yet the cost of computation is proportional only to the number of unknowns in the discretized equations. Until recently, most of the research in this area concentrated primarily on nested conforming finite element spaces. However, nonconforming finite elements are often used in applications because they are easier to program. In previous work, results on full W-cycle multigrid convergence for nonconforming finite elememts have been obtained by the principal investigator in the study of P1 nonconforming finite element applied to the Poisson equation and the Morley finite element applied to the biharmonic equation on polygonal domains. In this project, these results will be applied to the stationary Stokes equations and extended to domains with smooth boundary. Multigrid algorithms will be developed for nonconforming methods in linear elasticity and plate bending and also for mixed methods that are equivalent to nonconforming methods. The convergence of V-cycle nonconforming methods will be investigated. Numerical experiments will be performed for each of these problems.