An Eulerian finite element method for PDEs in time-dependent domains
An Eulerian finite element method for PDEs in time-dependent domains
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DOI：
10.1051/m2an/2018068
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
C. Lehrenfeld;M. Olshanskii
中科院分区：
文献类型：
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作者：
C. Lehrenfeld;M. Olshanskii
本文提出了一种新的求解演化区域上偏微分方程的有限元数值方法。该方法使用一个完全欧拉描述的域运动。物理域被嵌入到一个三角计算域中，可以以任意方式与时间无关的背景网格重叠。数值方法是基于时间导数的有限差分离散化和一个标准的几何不拟合有限元法，在空间域中的一个额外的稳定项。该方法的性能和分析依赖于定义在有界域上的函数在Sobolev空间中的基本扩展结果。本文包括一个完整的稳定性和误差分析，它占离散误差所造成的有限差分和有限元近似以及几何误差来自一个可能的近似恢复的物理域。几个数值例子说明了理论和证明了该方法的实际效率。
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.