A TRACE FINITE ELEMENT METHOD FOR PDES ON EVOLVING SURFACES
A TRACE FINITE ELEMENT METHOD FOR PDES ON EVOLVING SURFACES
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DOI：
10.1137/16m1099388
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发表时间：
2017-01-01
影响因子：
3.1
通讯作者：
Xu, Xianmin
中科院分区：
文献类型：
--
作者：
Olshanskii, Maxim A.;Xu, Xianmin
在本文中，我们提出了一种结合迹有限元方法和快速行进方法来求解演化曲面上的偏微分方程组的方法。数值方法基于曲面问题的欧拉描述，并采用与时间无关的背景网格，该网格不适合于曲面。例如，可以通过水平集方法隐含地给出表面及其演化。不需要将PDE从表面延伸出去。本文介绍的方法自然允许曲面经历拓扑变化和局部几何奇点。在最简单的情况下，数值方法在空间和时间上都是二阶精度的。高阶变种是可行的，但本文没有对其进行研究。我们给出了几个数值实验的结果，证明了该方法的收敛特性和处理具有拓扑变化的曲面的能力。
In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the Eulerian description of the surface problem and employs a time-independent background mesh that is not fitted to the surface. The surface and its evolution may be given implicitly, for example, by the level set method. Extension of the PDE off the surface is not required. The method introduced in this paper naturally allows a surface to undergo topological changes and experience local geometric singularities. In the simplest setting, the numerical method is second order accurate in space and time. Higher-order variants are feasible but not studied in this paper. We show results of several numerical experiments that demonstrate the convergence properties of the method and its ability to handle the case of the surface with topological changes.