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.
This work was supported by the Deutsche Forschungsgemeinschaft, SFB 123 and SFB 359 Universität Heidelberg
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References
Chen, H.; Rannacher, R.: Local error expansions and Richardson extrapolation for the streamline diffusion finite element method, 1993, to appear.
Eriksson, E.; Johnson, C.: Adaptive finite element methods for parabolic problems IV: Nonlinear problems, to appear.
Eriksson, E.; Johnson, C.; Thomée, V.: Time discretization of parabolic problems by the discontinuous Galerkin method, Math. Modelling and Numer. Anal., 19, 611643 (1985).
Hughes, T.J.R; Mallet, M; Mizukami, A: A new finite element formulation for computational fluid dynamics: II. Beyond SUPG, Comput. Methods Appl. Mech. Engrg. 54, 341–355 (1986).
Hughes, T.J.R; Mallet, M: A new finite element formulation for computational fluid dynamics: III. The general streamline operator for multidimensional aclvectivediffusive systems, Comput. Methods Appl. Mech. Engrg. 58, 305–328 (1986).
Hughes, T.J.R; Mallet, M: A new finite element formulation for computational fluid dynamics: IV. A discontinuity—capturing operator for multidimensional advectivediffusive systems, Comput. Methods Appl. Mech. Engrg. 58, 329–336 (1986).
Johnson, C.: Finite element methods for convection-diffusion problems, in: Computing Methods in Engineering and Applied Sciences V, (R. Glowinski and J.L. Lions, eds. ), North-Holland 1981.
Johnson, C.; Nävert, U.: An analysis of some finite element methods for advection-diffusion, in: Analytical and Numerical Approaches to Asymptotic Problems in Analysis, (O. Axelsson; L.S. Frank, and A. van der Sluis, eds. ), North-Holland 1981.
Johnson, C.; Nävert, U.; Pitkäranta, J.: Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45, 285–312 (1984).
Johnson, C.; Schatz, A.H.; Wahlbin, L.B.: Crosswind smear and pointwise errors in streamline diffusion finite element methods, Math. Comp., 49, 179, 25–38 (1987).
Johnson, C.; Szepessy, A.: On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp., 49, 180, 427–444 (1987).
Johnson, C.; Szepessy, A.: Adaptive finite element methods for conservation laws based on a posteriori error estimates, to appear.
Johnson, C.; Szepessy, A.; Hansbo, P.: On the convergence of shockcapturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54, 189, 107–129 (1990).
Nävert, U.: A finite element method for convection-diffusion problems, Thesis, Chalmers University of Technology, 1982 .
Rannacher, R.; Zhou, G.H.: Pointwise error estimate for nonstationary hyperbolic problems in the streamline diffusion method, 1992, to appear.
Rannacher, R.; Zhou, G.H.: Mesh orientation and refinement in the streamline diffusion method, 1993, to appear.
Smoller, J.: Shock Waves and Reaction—Diffusion Equations, Springer, Heidelberg, 1983.
Walter, A.: Ein Finite—Elemente—Verfahren zur numerischen Lösung von Erhaltungsgleichungen, Thesis, Universität Saarbrücken, 1988.
Walter, A.: Implementierung eines Finite-Elemente-Verfahrens für die Euler—Gleichungen der Gasdynamik, Preprint No. 500, SFB 123, Universität Heidelberg, 1989.
Zhou, G.H.: An adaptive streamline diffusion finite element method for hyperbolic systems in gas dynamics, Thesis, Universität Heidelberg, 1992.
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Rannacher, R., Zhou, G. (1994). Mesh Adaptation via a Predictor-Corrector Strategy in the Streamline Diffusion Method for Nonstationary Hyperbolic Systems. In: Hackbusch, W., Wittum, G. (eds) Adaptive Methods — Algorithms, Theory and Applications. Notes on Numerical Fluid Mechanics (NNFM). Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14246-1_16
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DOI: https://doi.org/10.1007/978-3-663-14246-1_16
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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