A TRACE FINITE ELEMENT METHOD FOR VECTOR-LAPLACIANS ON SURFACES
A TRACE FINITE ELEMENT METHOD FOR VECTOR-LAPLACIANS ON SURFACES
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DOI：
10.1137/17m1146038
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发表时间：
2018-01-01
影响因子：
2.9
通讯作者：
Reusken, Arnold
中科院分区：
文献类型：
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作者：
Gross, Sven;Jankuhn, Thomas;Reusken, Arnold
我们考虑了一个向量拉普拉斯问题的二维表面嵌入在一个三维区域，这是从表面流体的外部笛卡尔微分算子的基础上建模的结果。本文的主要议题是发展和分析的有限元方法的离散化这一表面偏微分方程。我们应用跟踪有限元技术，其中有限元空间的背景形状规则的四面体网格，是表面独立的用于离散化。为了满足解向量场与曲面相切的约束，我们引入了一个拉格朗日乘子。我们证明了所得到的鞍点公式的适定性。这个配方的离散变量，其中包含适当的稳定条款，是基于跟踪有限元空间。对于这种方法，我们得到最佳的离散误差界。进一步研究了离散鞍点问题的代数性质。特别是提出了一种最优Schur补预条件子。数值实验的结果。
We consider a vector-Laplace problem posed on a two-dimensional surface embedded in a three-dimensional domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included.