Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem
Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem
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广义 Robin 边值问题有限元逼近的适定性、正则性和收敛性分析
DOI：
10.1137/140954477
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Quarteroni
中科院分区：
文献类型：
--
作者：
Takahito Kashiwabara;C. M. Colciago;L. Dede’;A. Quarteroni
在本文中，我们通过引入具有$H^1(\Gamma)$迹的$H^1(\Omega)$函数的函数空间$H^1(\Omega; \Gamma)$，提出了对具有广义Robin边界条件的二阶偏微分方程进行数学和有限元分析，该二阶偏微分方程涉及Laplace--Beltrami算子，其中$\Gamma \subseteq \partial \Omega$。基于变分法，我们证明了广义Robin边值问题的解比标准Robin问题的解在边界上具有更好的正则性。我们通过有限元方法对广义 Robin 问题进行数值求解，目的是验证与空间 $H^1(\Omega; \Gamma)$ 相关的范数中误差的理论收敛率。
In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space $H^1(\Omega; \Gamma)$ of $H^1(\Omega)$-functions with $H^1(\Gamma)$-traces, where $\Gamma \subseteq \partial \Omega$. Based on a variational method, we prove that the solution of the generalized Robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard Robin problem. We numerically solve generalized Robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space $H^1(\Omega; \Gamma)$.