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arXiv · 1401.8214

Error analysis of a space-time finite element method for solving PDEs on evolving surfaces

Maxim A. Olshanskii·Arnold Reusken
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Abstract

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations posed on evolving hypersurfaces in $\mathbb{R}^d$, $d=2,3$. The method employs discontinuous piecewise linear in time -- continuous piecewise linear in space finite elements and is based on a space-time weak formulation of a surface PDE problem. Trial and test surface finite element spaces consist of traces of standard volumetric elements on a space-time manifold resulting from the evolution of a surface. We prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm. Furthermore, we derive regularity results for solutions of parabolic PDEs on an evolving surface, which we need in a duality argument used in the proof of the second order convergence estimate.

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Maxim A. Olshanskii, Arnold Reusken. 2014-04-09. Error analysis of a space-time finite element method for solving PDEs on evolving surfaces. https://arxiv.org/abs/1401.8214

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