Error Analysis of Ritz Finite Element Methods

在这个研究项目中，我们尝试发展了Ritz有限元方法的误差分析，并得到了以下结果。我们考虑了定义在二维欧氏空间中的共形映射到Jordan区域的有限元逼近。在分段线性有限元空间中定义了一组允许映象。然后，我们定义有限元(FE)共形映射为Dirichlet积分在可容许映射集中的极小值。在某些温和的假设下，我们证明了随着单位圆盘的三角剖分的细化，有限元共形映射收敛到精确的共形映射。我们还考虑了带角度的Jordan域。引入最优化理论中发展起来的“光滑化方法”，我们还可以计算到这些区域的有限元共形映射。文中给出了许多有趣的例子。三维欧氏空间中平均曲率在每个点上为常数的曲面称为H-曲面。我们考虑H-曲面的有限元逼近。在分段线性有限元空间中，有限元H-曲面被定义为能量泛函在一组允许映象中的驻点。证明了有限元H-曲面收敛到精确H-曲面。文中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.