表面Stokes问题的罚$H({\ m div})$协调有限元方法的误差分析

中文摘要 AI 辅助

近年来，针对表面Stokes方程已发展出多种有限元方法。为这一问题的速度-压力形式构造标准协调表面有限元方法，需要同时强制速度场的$H^1$协调性和切向性，而这在广义多面体逼近表面上（表面有限元通常在此类表面上提出）是不可能的。因此，针对该问题的方法要么弱强制速度场的切向性，要么弱强制其$H^1$协调性。本文基于经典的Brezzi-Douglas-Marini单元，为一种$H({\ m div})$协调方法提供了误差分析。在该方法中，速度场的连续性通过使用内罚型公式弱强制实现。该公式的优点包括精确强制不可压缩约束，以及其基于标准混合有限元对的构造。先前工作中对该方法的误差分析仅适用于最低阶单元，且未考虑“几何误差”，即由于用逼近曲面（有限元方法在其上提出）逼近偏微分方程所定义的连续曲面而导致的变分犯罪。本文在自然的间断Galerkin能量范数中提供了完整的误差分析，并计入了几何误差。同时给出了$L_2$速度误差的最优估计。在最低阶（多面体）曲面逼近情形下，我们证明最优$L_2$估计需要对网格节点相对于曲面的放置施加比通常更严格的条件。数值实验用于说明我们的结果。

英文摘要

In recent years a number of finite element methods have been developed for the surface Stokes equations. Constructing standard conforming surface finite element methods for the velocity-pressure formulation of this problem would require simultaneously enforcing $H^1$ conformity and tangentiality of the velocity field, which is not possible on the generalized polyhedral approximating surfaces on which surface FEM are typically posed. Thus methods for this problem weakly enforce either tangentiality or $H^1$ conformity of the velocity field. In this paper we provide an error analysis for an $H({\rm div})$-conforming method based on classical Brezzi-Douglas-Marini elements. In this method continuity of the velocity field is weakly enforced using an interior penalty-type formulation. Advantages of this formulation include exact enforcement of the incompressibility constraint and its construction based on a standard mixed finite element pair. Error analysis of this method in previous work was only valid for lowest-order elements and did not account for ``geometric errors'', i.e., variational crimes due to approximation of the continuous surface on which the PDE is posed by an approximating surface on which the finite element method is posed. Here we provide a full error analysis in a natural discontinuous Galerkin energy norm, with geometric errors accounted for. Optimal estimates for the $L_2$ velocity error are also presented. In the case of lowest-order (polyhedral) surface approximations, our proof of optimal $L_2$ estimates requires a stricter than usual condition on the placement of mesh nodes relative to the surface. Numerical experiments are used to illustrate our results.

发表机构

• Texas A&M University（德克萨斯农工大学）
• Istanbul Bilgi University（伊斯坦布尔比尔吉大学）

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