Zhiming Chen (Caltech)
Abstract:
Adaptive finite element method based on a posteriori error estimates has attracted ever increasing interests among mathematical and engineering communities. This method provides a systematic way to resolve the singularities of practical problems but still keep the total computational costs in minimum via locally refining and coarsening the underlying finite element meshes. This property is particularly important when solving nonlinear problems since it is often for nonlinear PDEs the singularities are coming from the nonlinear nature of the problems which are a priori unknown, rather from the rough coefficients of equations or from the geometry of the underlying domains as in the case of linear elliptic and parabolic equations. In this talk I shall report our recent efforts in solving the time dependent Ginzburg-Landau model in superconductivity (joint work with S. Dai) and the continuous casting Stefan problem (joint work with R.H. Nochetto and A. Schmidt) by using adaptive finite element methods. In both cases the key issue is to derive reliable and efficient a posteriori error estimates. Numerical tests will be presented.