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Crosswind smear and pointwise errors in streamline diffusion finite element methods

Authors: C. Johnson, A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 49 (1987), 25-38
MSC: Primary 65N30; Secondary 35B25
MathSciNet review: 890252
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Abstract: For a model convection-dominated singularly perturbed convection-diffusion problem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost $ {h^{5/4}}$ under local smoothness assumptions.

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  • [1] O. Axelsson & W. Layton, Defect Correction Methods for Convection Dominated Convection-Diffusion Problems, Technical report, University of Nijmegen. (To appear.) MR 1070965 (92a:65272)
  • [2] A. N. Brooks & T. J. R. Hughes, "Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations," Comput. Methods Appl. Mech. Engrg., v. 32, 1982, pp. 199-259. MR 679322 (83k:76005)
  • [3] J. Douglas, Jr., T. Dupont & L. B. Wahlbin, "The stability in $ {L^q}$ of the $ {L^2}$ projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193-197. MR 0388799 (52:9633)
  • [4] W. Eckhaus, "Boundary layers in linear elliptic singular perturbation problems," SIAM Rev., v. 14, 1972, pp. 225-270. MR 0600325 (58:29072)
  • [5] W. Eckhaus & E. M. De Jager, "Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type," Arch. Rational Mech. Anal., v. 23, 1966, pp. 26-86. MR 0206464 (34:6283)
  • [6] R. Gore, "The dead do tell tales at Vesuvius," National Geographic, v. 165, no. 5 (May 1984), pp. 557-613.
  • [7] T. J. R. Hughes & A. N. Brooks, "A multidimensional upwind scheme with no crosswind diffusion," in Finite Element Methods for Convection Dominated Flows (T. J. R. Hughes, ed.), AMD, Vol. 34, ASME, New York, N. Y., 1979, pp. 19-35. MR 571681 (81f:76040)
  • [8] C. Johnson & U. Nävert, An analysis of some finite element methods for advection-diffusion problems," in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (O. Axelsson, L. S. Frank, and A. van der Sluis, eds.), North-Holland, Amsterdam, 1981, pp. 99-116. MR 605502 (82e:65127)
  • [9] C. Johnson, U. Nävert, & J. Pitkäranta, "Finite element methods for linear hyperbolic problems," Comput. Methods Appl. Mech. Engrg., v. 45, 1984, pp. 285-312. MR 759811 (86a:65103)
  • [10] C. Johnson & J. Saranen, "Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations," Math. Comp., v. 47, 1986, pp. 1-18. MR 842120 (88b:65133)
  • [11] J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Springer-Verlag, Berlin and New York, 1973. MR 0600331 (58:29078)
  • [12] U. Nävert, A Finite Element Method for Convection-Diffusion Problems, Thesis, Chalmers University of Technology and University of Gothenburg, 1982.
  • [13] J. Pitkäranta, Personal communication on numerical experiments.
  • [14] G. D. Raithby & K. E. Torrance, "Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow," Comput. & Fluids, v. 2, 1974, pp. 191-206. MR 0345431 (49:10167)
  • [15] L. B. Wahlbin, "A dissipative Galerkin method for the numerical solution of first order hyperbolic equations," in Mathematical Aspects of Finite Elements in Partial Differential Equations (C. deBoor, ed.), Academic Press, New York, 1974, pp. 147-169. MR 0658322 (58:31929)

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Keywords: Finite element method, convection dominated
Article copyright: © Copyright 1987 American Mathematical Society

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