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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 under local smoothness assumptions.

**[1]**O. Axelsson and W. Layton,*Defect correction methods for convection-dominated convection-diffusion problems*, RAIRO Modél. Math. Anal. Numér.**24**(1990), no. 4, 423–455 (English, with French summary). MR**1070965****[2]**Alexander N. Brooks and Thomas 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.**32**(1982), no. 1-3, 199–259. FENOMECH ’81, Part I (Stuttgart, 1981). MR**679322**, 10.1016/0045-7825(82)90071-8**[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****[4]**Wiktor Eckhaus,*Boundary layers in linear elliptic singular perturbation problems*, SIAM Rev.**14**(1972), 225–270. MR**0600325****[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****[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 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****[8]**Claes Johnson and Uno Nävert,*An analysis of some finite element methods for advection-diffusion problems*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 99–116. MR**605502****[9]**Claes Johnson, Uno Nävert, and Juhani Pitkäranta,*Finite element methods for linear hyperbolic problems*, Comput. Methods Appl. Mech. Engrg.**45**(1984), no. 1-3, 285–312. MR**759811**, 10.1016/0045-7825(84)90158-0**[10]**Claes Johnson and Jukka Saranen,*Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations*, Math. Comp.**47**(1986), no. 175, 1–18. MR**842120**, 10.1090/S0025-5718-1986-0842120-4**[11]**J.-L. Lions,*Perturbations singulières dans les problèmes aux limites et en contrôle optimal*, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR**0600331****[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 and K. E. Torrance,*Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow*, Internat. J. Comput. & Fluids**2**(1974), 191–206. MR**0345431****[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 Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. Publication No. 33. MR**0658322**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
35B25

Retrieve articles in all journals with MSC: 65N30, 35B25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890252-8

Keywords:
Finite element method,
convection dominated

Article copyright:
© Copyright 1987
American Mathematical Society