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\begin{abstract}
Many physical processes and biological phenomena could be modeled by partial differential equations on surfaces or manifolds. Recently, the method to solve surface PDEs has arisen much interests in the community of numerical analysis. In this talk, I will introduce some trace finite element methods for convection-diffusion equations on evolving surfaces. The finite element space is a trace of a standard finite element space defined in the neighboring region of the surface. To deal with the evolving surface, an extension of the solution is done by fast marching method or a stabilization technique. The methods are based on a Eulerian framework and easily treat shape and topology changes of the surface. Numerical experiments and error estimates show the optimal convergence of the methods.
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