Finite element analysis for a coupled bulk-surface partial differential equation
Finite element analysis for a coupled bulk-surface partial differential equation
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DOI：
10.1093/imanum/drs022
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发表时间：
2013-04-01
影响因子：
2.1
通讯作者：
Ranner, Thomas
中科院分区：
文献类型：
--
作者：
Elliott, Charles M.;Ranner, Thomas
在本文中，我们定义了一个新的有限元方法的数值逼近的解决方案的偏微分方程在一个散装区域耦合的表面偏微分方程的边界上的散装域。其关键思想是采取多面体近似的整体区域组成的联盟的单纯形，并使用分段多项式边界面作为近似的表面。定义了两个有限元空间，一个在体区域和一个在表面上，通过采取所有连续函数的集合，这些连续函数在每个体单形或边界面上也是分段多项式。我们研究这种方法的背景下，一个模型椭圆问题，特别是，我们看看适定性的系统使用变分制剂，推导出扰动估计所产生的域近似和应用这些找到最优阶误差估计。一个数值实验，证明了收敛的顺序。
In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.