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arXiv · 1304.6155

An Eulerian space-time finite element method for diffusion problems on evolving surfaces

Maxim A. Olshanskii·Arnold Reusken·Xianmin Xu
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Abstract

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum $\Bbb{R}^{d+1}$. 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 non-standard, (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.

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Maxim A. Olshanskii, Arnold Reusken, Xianmin Xu. 2013-04-23. An Eulerian space-time finite element method for diffusion problems on evolving surfaces. https://arxiv.org/abs/1304.6155

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