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), 2538
MSC:
Primary 65N30; Secondary 35B25
MathSciNet review:
890252
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Abstract: For a model convectiondominated singularly perturbed convectiondiffusion 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 under local smoothness assumptions.
 [1]
O.
Axelsson and W.
Layton, Defect correction methods for convectiondominated
convectiondiffusion problems, RAIRO Modél. Math. Anal.
Numér. 24 (1990), no. 4, 423–455
(English, with French summary). MR 1070965
(92a:65272)
 [2]
Alexander
N. Brooks and Thomas
J. R. Hughes, Streamline upwind/PetrovGalerkin formulations for
convection dominated flows with particular emphasis on the incompressible
NavierStokes equations, Comput. Methods Appl. Mech. Engrg.
32 (1982), no. 13, 199–259. FENOMECH
’81, Part I (Stuttgart, 1981). MR 679322
(83k:76005), http://dx.doi.org/10.1016/00457825(82)900718
 [3]
Lars
Wahlbin, Error estimates for a Galerkin method for a class of model
equations for long waves, Numer. Math. 23 (1975),
289–303. With an appendix by Lars Wahlbin, Jim Douglas, Jr. and Todd
Dupont. MR
0388799 (52 #9633)
 [4]
Wiktor
Eckhaus, Boundary layers in linear elliptic singular perturbation
problems, SIAM Rev. 14 (1972), 225–270. MR 0600325
(58 #29072)
 [5]
W.
Eckhaus and E.
M. de Jager, Asymptotic solutions of singular perturbation problems
for linear differential equations of elliptic type, Arch. Rational
Mech. Anal. 23 (1966), 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. 557613.
 [7]
T.
J. R. Hughes and A.
Brooks, A multidimensional upwind scheme with no crosswind
diffusion, Finite element methods for convection dominated flows
(Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979)
AMD, vol. 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979,
pp. 19–35. MR 571681
(81f:76040)
 [8]
Claes
Johnson and Uno
Nävert, An analysis of some finite element methods for
advectiondiffusion problems, Analytical and numerical approaches to
asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen,
1980) NorthHolland Math. Stud., vol. 47, NorthHolland,
AmsterdamNew York, 1981, pp. 99–116. MR 605502
(82e:65127)
 [9]
Claes
Johnson, Uno
Nävert, and Juhani
Pitkäranta, Finite element methods for linear hyperbolic
problems, Comput. Methods Appl. Mech. Engrg. 45
(1984), no. 13, 285–312. MR 759811
(86a:65103), http://dx.doi.org/10.1016/00457825(84)901580
 [10]
Claes
Johnson and Jukka
Saranen, Streamline diffusion methods for the
incompressible Euler and NavierStokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120
(88b:65133), http://dx.doi.org/10.1090/S00255718198608421204
 [11]
J.L.
Lions, Perturbations singulières dans les problèmes
aux limites et en contrôle optimal, Lecture Notes in
Mathematics, Vol. 323, SpringerVerlag, BerlinNew York, 1973 (French). MR 0600331
(58 #29078)
 [12]
U. Nävert, A Finite Element Method for ConvectionDiffusion Problems, Thesis, Chalmers University of Technology and University of Gothenburg, 1982.
 [13]
J. Pitkäranta, Personal communication on numerical experiments.
 [14]
G.
D. Raithby and K.
E. Torrance, Upstreamweighted differencing schemes and their
application to elliptic problems involving fluid flow, Internat. J.
Comput.\thinspace&\thinspace Fluids 2 (1974),
191–206. MR 0345431
(49 #10167)
 [15]
Lars
B. Wahlbin, A dissipative Galerkin method for the numerical
solution of first order hyperbolic equations, Mathematical aspects of
finite elements in partial differential equations (Proc. Sympos., Math.
Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center,
Univ. of WisconsinMadison, Academic Press, New York, 1974,
pp. 147–169. Publication No. 33. MR 0658322
(58 #31929)
 [1]
 O. Axelsson & W. Layton, Defect Correction Methods for Convection Dominated ConvectionDiffusion Problems, Technical report, University of Nijmegen. (To appear.) MR 1070965 (92a:65272)
 [2]
 A. N. Brooks & T. J. R. Hughes, "Streamline upwind PetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations," Comput. Methods Appl. Mech. Engrg., v. 32, 1982, pp. 199259. MR 679322 (83k:76005)
 [3]
 J. Douglas, Jr., T. Dupont & L. B. Wahlbin, "The stability in of the projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193197. MR 0388799 (52:9633)
 [4]
 W. Eckhaus, "Boundary layers in linear elliptic singular perturbation problems," SIAM Rev., v. 14, 1972, pp. 225270. 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. 2686. MR 0206464 (34:6283)
 [6]
 R. Gore, "The dead do tell tales at Vesuvius," National Geographic, v. 165, no. 5 (May 1984), pp. 557613.
 [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. 1935. MR 571681 (81f:76040)
 [8]
 C. Johnson & U. Nävert, An analysis of some finite element methods for advectiondiffusion problems," in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (O. Axelsson, L. S. Frank, and A. van der Sluis, eds.), NorthHolland, Amsterdam, 1981, pp. 99116. 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. 285312. MR 759811 (86a:65103)
 [10]
 C. Johnson & J. Saranen, "Streamline diffusion methods for the incompressible Euler and NavierStokes equations," Math. Comp., v. 47, 1986, pp. 118. MR 842120 (88b:65133)
 [11]
 J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, SpringerVerlag, Berlin and New York, 1973. MR 0600331 (58:29078)
 [12]
 U. Nävert, A Finite Element Method for ConvectionDiffusion 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, "Upstreamweighted differencing schemes and their application to elliptic problems involving fluid flow," Comput. & Fluids, v. 2, 1974, pp. 191206. 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. 147169. MR 0658322 (58:31929)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708902528
PII:
S 00255718(1987)08902528
Keywords:
Finite element method,
convection dominated
Article copyright:
© Copyright 1987
American Mathematical Society
