ERROR ANALYSIS OF A SPACE-TIME FINITE ELEMENT METHOD FOR SOLVING PDES ON EVOLVING SURFACES~*

摘要

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations (PDEs) posed on evolving hypersurfaces in ?~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.