Summary
The Galerkin /least-squares finite element method is considered as a tool for solving singularly perturbed partial differential equations of elliptic type on adaptively refined grids. Local error estimates in subdomains away from boundary and interior layers are uniformly valid with respect to the small parameter. Boundary and interior layers can be resolved using locally anisotropic mesh refinement. Numerical examples are given for scalar convection-diffusion equations and for incompressible Navier—Stokes flow problems.
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References
Auge,A., Lube,G.: Regularized Mixed Finite Element Approximations of Incompressible Flow Problems. Preprint TU Magdeburg 1993.
Behr,M.: Stabilized Finite Element Methods for Incompressible Flows with Emphasis on Moving Boundaries and Interfaces. Thesis, Univ. of Minnesota 1992.
Guo,W., Stynes,M.: Pointwise Error Estimates for a Streamline Diffusion Scheme on a Shishkin Mesh for a Convection-Diffusion Problem. Preprint 1993–2, Dept. of Math., Univ. College. Cork, Ireland 1993.
EJ] Eriksson,K., Johnson,C.: Adaptive Streamline Diffusion Finite Element Methods for Convection-Diffusion Problems. Math. Comp.. (to appear).
Franca, L. P., Frey, S. L., Hughes, T. J. R.: Stabilized finite element methods. Part I. Comp. Meths. Appl. Mech. Engrg. 95 (1992), 253–276.
Johnson, C., Schatz, A. H., Wahlbin, L. B.: Crosswind Smear and Pointwise Errrors in Streamline Diffusion Finite Element Methods. Math. Comp. 49 (1987) 25–38.
Kornhuber,R., Roitzsch,R.: On Adaptive Grid Refinement in the Presence of Internal or Boundary Layers. Preprint SC 89–5, K.Zuse-Zentrum Berlin (1989).
Lang,J.: An Adaptive Finite Element Method for Convec- tion-Diffusion Problems by Interpolation Techniques. Technical Report TR 91–4, K.Zuse-Zentrum Berlin 1991.
Lube,G.: Stabilized Galerkin Finite Element Methods for Convection Dominated and Incompressible Flow Problems. Proc. 37. Sem. Numer. Analysis and Math. Modell. 1991. Banach Center Public. Warsaw 1993.
Nävert,U.: A Finite Element Method for Convection-Diffusion Problems. Thesis, Chalmers Univ. Göteborg 1982
Shakib,F.: Finite Element Analysis of the Compressible Euler and Navier-Stokes Equations. Ph.D.Thesis Stan- ford Univ. 1989.
Shishkin,G. I.: Approximation of the solution of singularly perturbed boundary value problems with a parabolic boundary layer (Russ.). Sh.Vytsch.Matem. Mat.Fis. 29 (1989) 7, 963–977.
WeiB,D.: Feldberechnungen mit der FEM zur Simulation von Temperatur- und Gefügeverteilungen bei Schweißprozessen. Forschungsbericht TU Magdeburg, GKMBI 1992
Zhou,G.: An Adaptive Streamline Diffusion Finite Element Method for Hyperbolic Systems in Gas Dynamics. Thesis, Univ. Heidelberg 1992.
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© 1994 Springer Fachmedien Wiesbaden
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Auge, A., Lube, G., Weiß, D. (1994). Galerkin/Least-Squares-FEM and Anisotropic Mesh Refinement. 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_1
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DOI: https://doi.org/10.1007/978-3-663-14246-1_1
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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