An adaptive finite element method for unsteady convection-dominated flows with stiff source terms

This paper considers nonsteady convection-dominated flows with stiff source terms. As a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically. It has good shock-capturing features, is implicit and L-stable. Moreover, being a Galerkin method, it admits residual-based weighted a posteriori error estimates of optimal order. We avoid the use of global stability constants by solving the dual problem explicitly. This in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used to optimize the resulting adaptive scheme. All theoretical results are substantiated by numerical test examples. For a physical application, we turn to detonation waves at moderate relaxation times in one space dimension. The finite element method reproduces the Chapman Jouguet speed of detonations and their wave structure well. Our error estimator proves accurate and sensitive to parameters, at least on time intervals of moderate length. The adaptive scheme based on this estimator produces fairly accurate results that appear to be the best available with the present strategy. All ingredients of the numerical scheme can be extended in a natural way to higher space dimensions. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(97-15) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)DEGerman