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

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