Mesh adaptation via a predictor-corrector strategy in the streamline diffusion method for nonstationary hyperbolic systems. (English) Zbl 0808.65098

Rannacher, Rolf; Zhou, Guohui

Hackbusch, Wolfgang (ed.) et al., Adaptive methods - algorithms, theory and applications. Proceedings of the 9th GAMM-Seminar Kiel, Germany, January 22-24, 1993. Braunschweig: Vieweg. Notes Numer. Fluid Mech. 46, 236-250 (1994).

Summary: Streamline diffusion is a well-known damping strategy in the numerical approximation of transport-dominated transport-diffusion problems by the finite element method. It combines the effect of higher-order upwinding with the flexibility of the variational approach. However, the reliable and accurate resolution of shock-like solution structures requires a refinement of the computational mesh in space and also in time.
In this note we propose a strategy for such a mesh refinement within an implicit time stepping process which is based on a local predictor- corrector concept. This approach allows for significant savings with respect to storage and computing time. Some test results are reported for the 1-d Riemann shock tube problem. The mechanism underlying this mesh control strategy can be explained through a rigorous theoretical analysis.
For the entire collection see [Zbl 0802.00027].

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50	Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35L70	Second-order nonlinear hyperbolic equations

Keywords:

streamline diffusion; transport-diffusion problems; finite element method; upwinding; shock-like solution; mesh refinement; implicit time stepping process; predictor-corrector concept; Riemann shock tube problem; mesh control strategy