Abstract
Much work has been done upon the fast solution of the linear systems arising from upwind difference and central difference approximations to convection dominated convection diffusion equations and yet neither scheme produces acceptable approximate solutions to the original boundary value problem. Still, if these two discretizations are combined via a defect correction approach one obtains an algorithm with many of the best features of both and for which the insights on the fast solution of the associated linear system are very useful. In this report, we describe the algorithm and summarize some of the authors' recent work which gives local error estimates in 2-D on a finite element implementation of this promising scheme.
Key Works
- Convection-diffusion equation
- defect correction method
- interior estimates
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References
O. Axelsson, On the numerical solution of convection dominated, convection-diffusion problems, in: Math. Meth. Energy Res. (K.I. Gross, e.d.), SIAM, Philadelpha, 1984.
O. Axelsson, Stability and error estimates of Galerkin finite element aproximations for convection-diffusion equations, I.M.A. Journal Numerical Analysis, 1 (1981), 329–345.
O. Axelsson and W. Layton, Defect correction methods for convection dominated, convection-diffusion problems, M.A.N. (formerly R.A.I.R.O. J. Numer. Anal.), 24 (1990), 423–455.
W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems, SIAM Review, 14 (1972), 225–270.
V. Ervin and W. Layton, High resolution minimal storage algorithms for convection dominated, convection diffusion equations, pp. 1173–1201 in: Trans. of the Fourth Army Conf.on Appl. Math. and Comp., 1987.
V. Ervin and W. Layton, An analysis of a defect correction method for a model convection diffusion equation, SIAM J. Numer. Anal., 26 (1989), 169–179.
P. W. Hemker, Mixed defect correction iteration for the accurate solution of the convecton diffusion equations, pp. 485–501 in: Multigrid Methods, L.N.M. vol. 960, (W. Hackbusch and U. Trottenberg, eds.) Springer Verlag, Berlin 1982.
P. W. Hemker, The use of defect correction for the solution of a singularly perturbed o.d.e., preprint.
C. Johnson and U. Navert, An analysis of some finite element methods for advection diffusion problems, in: Anal. and Numer. Approaches to Asym. Probs. in Analysis (O. Axelsson, L. S. Frank and A. van der Sluis, eds.) North Holland, 1981, 99–116.
C. Johnson, U. Navert and J. Pitkaranta, Finite element methods for linear hyperbolic problems, Comp. Meth. Appl. Mech. Eng., 45 (1984), 285–312.
C. Johnson, A. H. Schatz and L. B. Wahlbin, Crosswind smear and pointwise errors in streamline diffusion finite element methods, Math. Comp., 49 (1987), 25–38.
U. Navert, A finite element method for convection diffusion problems, Thesis, Chalmers Inst. of Tech., 1982.
O. Axelsson and W. Layton, Optimal interior estimates for the defect-correction, finite element approach to 2-D convection-diffusion problems, ICMA report 88–116, Univ. of Pittsburgh, 1988.
V. Pereyra, Iterated defect corrections for nonlinear operator equations, Num. Math., 10 (1967) 316–323.
O. Axelsson, V. Eijkhout, B. Polman and P. Vassilevski, Incomplete block-matrix factorization iterative methods for convection-diffusion problems, BIT, to appear.
J. Boland and W. Layton, Error analysis of finite element methods for steady natural convection problems, to appear in: Numerical Functional Anal. and Opt.
P. W. Hemker and S. P. Spekreijse, Multigrid solution of the steady Euler equations, CWI report NM-R8507, (to appear in Appl. Num. Math.), 1985.
S. P. Spekreijse, Second order accurate upwind solutions of the 2-D steady Euler equations by use of a defect correction method, CWI Report NM-R8520, (to appear in: Proc. 2nd European Multigrid Conf.), 1985.
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© 1990 Springer-Verlag
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Axelsson, O., Layton, W. (1990). Iteration method as discretization procedures. In: Axelsson, O., Kolotilina, L.Y. (eds) Preconditioned Conjugate Gradient Methods. Lecture Notes in Mathematics, vol 1457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090908
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DOI: https://doi.org/10.1007/BFb0090908
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