论文标题
分析表面矢量 - 拉普拉斯本本特征的有限元方法
Analysis of finite element methods for surface vector-Laplace eigenproblems
论文作者
论文摘要
在本文中，我们研究了表面矢量 - 拉普拉斯本本特征问题的有限元离散。我们考虑两种已知类别的有限元方法，即基于Dziuk-Elliott表面有限元方法的向量类似物，另一种基于所谓的痕量有限元技术。两类方法中的一个关键成分是一种惩罚方法，用于在薄弱的意义上实现向量场的切向性。这种惩罚和由表面的数值近似产生的扰动导致了矢量 - 宽面特征特征性的变异公式的离散化的基本不符合性。我们提出了一个普遍的抽象框架，适用于本本本特征的这种不合格的离散化。特征值和特征向量近似的误差界限都取决于某些一致性和近似性参数。讨论了这些边界的清晰度。数值实验的结果说明了表面矢量 - 拉普拉斯本本特征问题的这种有限元离散的某些收敛性。
In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.
