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In this paper, we study numerical methods for the solution of partialdifferential equations on evolving surfaces. The evolving hypersurface in$\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-timecontinuum $\Bbb{R}^{d+1}$. We derive and analyze a variational formulation fora class of diffusion problems on the space-time manifold. For this variationalformulation new well-posedness and stability results are derived. The analysisis based on an inf-sup condition and involves some natural, but non-standard,(anisotropic) function spaces. Based on this formulation a discrete in timevariational formulation is introduced that is very suitable as a starting pointfor a discontinuous Galerkin (DG) space-time finite element discretization.This DG space-time method is explained and results of numerical experiments arepresented that illustrate its properties.

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