SCAIM Seminar: Ye Yang

Abstract:

There is a continuing interest in developing numerical methods for treating problems that are characterized by multiple time scale features. Traditional finite element method (FEM) based on semi-discrete schemes, however, is not well suited for these classes of problems due to their lack of flexibility in establishing multiscale approximations in the temporal domain. In this presentation, we show that a multiscale method that is capable of incorporating both multiple spatial and temporal features can be established based on the space-time discontinuous Galerkin method which was originally developed in the context of linear elastodynamics. After an initial assessment of the convergence and its connection to the various time stepping algorithms, we show that space-time FEM is a stable, high-order convergent numerical method. We further explore the incorporation of fine scale features based on the extended finite element method. The nonlinear formulation incorporating enriched space-time FEM with stabilization least-square term is further developed and numerical solution based on GMRES is proposed. Through numerical examples, it is shown that multiscale space-time FEM enjoys superior convergence properties over the traditional space-time FEM and the proposed method represent a new paradigm towards resolving structural and solid mechanics problems with strong temporal nonlinearity.