Error Analysis of Ritz Finite Element Methods

在这个研究项目中，我们尝试开发了一种Ritz有限元方法的误差分析，并获得了以下结果。我们考虑了二维欧几里德空间中单位圆盘到Jordan域的共形映射的有限元逼近。在分段线性有限元空间中定义了可容许映射集。然后将有限元共形映射定义为可容许映射集合中Dirichlet积分的最小值。在某些温和的假设下，我们证明了当单位圆盘的三角剖分越来越精细时，有限元共形映射收敛于一个精确的共形映射。我们还考虑了带角度的约旦域。引入最优化理论中发展出来的“平滑法”，我们也可以计算这些域的有限元共形映射。文中给出了许多有趣的例子。在三维欧几里德空间中，其每一点的平均曲率为常数的曲面称为h曲面。考虑h曲面的有限元逼近。将有限元h曲面定义为分段线性有限元空间中某一组可容许映射中的能量泛函的驻点。证明了有限元h曲面收敛于精确h曲面。给出了许多数值算例。

In this research project, we have tried to develop an error analysis of Ritz finite element methods, and have obtained the following results.We have considered finite element approximations of conformal mappings from the unit disk to Jordan domains defined in two-dimensional Euclidean space. The set of admissible mappings is defined in piecewise linear finite element space. We then define the finite element (FE) conformal mappings as the minimizer of the Dirichlet integral in the set of admissible mappings. Under certain mild assumptions we have shown FE conformal mappings converge to a exact conformal mapping as triangulation of the unit disk is getting refined. We also considered Jordan domains with angles. Introducing "the smoothing method", developed in optimization theory, we may also compute FE conformal mappings to those domains. Many interesting examples are given.Surfaces in three-dimensional Euclidean space on which the mean curvature is constant at every point are called H-surfaces. We consider finite element approximation of H-surfaces. Finite element H-surfaces are defined as stationary point of an energy functional in a certain set of admissible mappings in the piecewise linear finite element space. It is proven that finite element H-surfaces converge to an exact H-surface. Many numerical examples are given.