Defect corrections for multigrid solutions of the Dirichlet problem in general domains
HTML articles powered by AMS MathViewer
- by Winfried Auzinger PDF
- Math. Comp. 48 (1987), 471-484 Request permission
Abstract:
Recently, the technique of defect correction for the refinement of discrete solutions to elliptic boundary value problems has gained new acceptance in connection with the multigrid approach. In the present paper we give an analysis of a specific application, namely to finite-difference analogues of the Dirichlet problem for Helmholtz’s equation, emphasizing the case of nonrectangular domains. A quantitative convergence proof is presented for a class of convex polygonal domains.References
-
W. Auzinger, Defektkorrektur für Diskretisierungen des Dirichlet-Problems in allgemeinen Gebieten, Ph. D. Thesis, Technical University of Vienna, Oct., 1984.
W. Auzinger, Defect Corrections for Multigrid Solutions of the Dirichlet Problem in General Domains, Report 61/85, Inst. f. Angewandte und Numerische Mathematik, Technical University of Vienna, 1985.
W. Auzinger, DCMG01: A Multigrid Code with Defect Correction to Solve $\Delta U - c(x,y)U = f(x,y)\;(on\;\Omega )$, $U = g(x,y)\;(on\;\partial \Omega )$, on Nonrectangular Bounded Domains $\Omega$ with High Accuracy, Arbeitspapier der GMD, Nr. 127, St. Augustin, Germany, January, 1985.
- W. Auzinger and H. J. Stetter, Defect corrections and multigrid iterations, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 327–351. MR 685777
- Dietrich Braess, The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation, Math. Comp. 42 (1984), no. 166, 505–519. MR 736449, DOI 10.1090/S0025-5718-1984-0736449-6
- J. H. Bramble and B. E. Hubbard, On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation, Numer. Math. 4 (1962), 313–327. MR 149672, DOI 10.1007/BF01386325
- R. Frank, J. Hertling, and J. P. Monnet, The application of iterated defect correction to variational methods for elliptic boundary value problems, Computing 30 (1983), no. 2, 121–135 (English, with German summary). MR 698124, DOI 10.1007/BF02280783
- Wolfgang Hackbusch, Convergence of multigrid iterations applied to difference equations, Math. Comp. 34 (1980), no. 150, 425–440. MR 559194, DOI 10.1090/S0025-5718-1980-0559194-5
- W. Hackbusch, Multigrid convergence theory, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 177–219. MR 685774
- W. Hackbusch, On multigrid iterations with defect correction, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 461–473. MR 685783
- Wolfgang Hackbusch, On the regularity of difference schemes. II. Regularity estimates for linear and nonlinear problems, Ark. Mat. 21 (1983), no. 1, 3–28. MR 706637, DOI 10.1007/BF02384298
- Hans J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1977/78), no. 4, 425–443. MR 474803, DOI 10.1007/BF01432879
- Klaus Stüben and Ulrich Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 1–176. MR 685773
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 471-484
- MSC: Primary 65B05; Secondary 65N20
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878685-7
- MathSciNet review: 878685