Some aspects of adaptive grid technology related to boundary and interior layers
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Journal of Computational Physics
On Adaptive Eulerian---Lagrangian Method for Linear Convection---Diffusion Problems
Journal of Scientific Computing
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We consider nonstationary 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 and 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 actually solving the dual problem. This in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing 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 well reproduces the Chapman Jouguet speed of detonations and their wave structure. 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 which 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.