AN EULERIAN SPACE-TIME FINITE ELEMENT METHOD FOR DIFFUSION PROBLEMS ON EVOLVING SURFACES
AN EULERIAN SPACE-TIME FINITE ELEMENT METHOD FOR DIFFUSION PROBLEMS ON EVOLVING SURFACES
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DOI：
10.1137/130918149
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发表时间：
2014-01-01
影响因子：
2.9
通讯作者：
Xu, Xianmin
中科院分区：
文献类型：
--
作者：
Olshanskii, Maxim A.;Reusken, Arnold;Xu, Xianmin
本文研究了演化曲面上偏微分方程的数值解法。R-d中的演化超曲面在时空连续体Rd+1中定义了一个d维时空流形。导出并分析了一类时空流形上扩散问题的变分公式。对于该变分公式，得到了新的适定性和稳定性结果。该分析是基于一个相互依存的条件，并涉及一些自然的，但非标准的（各向异性）函数空间。在此基础上，引入了一个离散的时变公式，它非常适合作为不连续伽辽金时空有限元离散化的起始点。对该方法进行了解释，并给出了数值实验结果，说明了该方法的性质。
In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in R-d defines a d-dimensional space-time manifold in the space-time continuum Rd+1. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but nonstandard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.