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Defect corrections for multigrid solutions of the Dirichlet problem in general domains

Author: Winfried Auzinger
Journal: Math. Comp. 48 (1987), 471-484
MSC: Primary 65B05; Secondary 65N20
MathSciNet review: 878685
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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.

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  • [1] W. Auzinger, Defektkorrektur für Diskretisierungen des Dirichlet-Problems in allgemeinen Gebieten, Ph. D. Thesis, Technical University of Vienna, Oct., 1984.
  • [2] 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.
  • [3] 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.
  • [4] W. Auzinger & H. J. Stetter, Defect Corrections and Multigrid Iterations, in Lecture Notes in Math., vol. 960, Springer-Verlag, Berlin and New York, 1982, pp. 327-351. MR 685777 (84d:65078)
  • [5] D. Braess, "The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation," Math. Comp., v. 42, 1984, pp. 505-519. MR 736449 (85f:65105)
  • [6] J. H. Bramble & B. E. Hubbard, "On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation," Numer. Math., v. 4, 1962, pp. 313-327. MR 0149672 (26:7157)
  • [7] R. Frank, J. Hertling & J. P. Monnet, "The application of iterated defect correction to variational methods for elliptic boundary value problems," Computing, v. 30, 1983, pp. 121-135. MR 698124 (84d:65081)
  • [8] W. Hackbusch, "Convergence of multi-grid iterations applied to difference equations," Math. Comp., v. 34, 1980, pp. 425-440. MR 559194 (83b:65112)
  • [9] W. Hackbusch, Multigrid Convergence Theory, in Lecture Notes in Math., vol. 960, Springer-Verlag, Berlin and New York, 1982, pp. 177-219. MR 685774 (84k:65113)
  • [10] W. Hackbusch, On Multigrid Iterations with Defect Corrections, in Lecture Notes in Math., vol. 960, Springer-Verlag, Berlin and New York, 1982, pp. 461-473. MR 685783 (84c:65143)
  • [11] W. Hackbusch, "On the regularity of difference schemes. Part II: Regularity estimates for linear and nonlinear problems," Ark. Mat., v. 21, 1983, pp. 1-28. MR 706637 (85h:65220)
  • [12] H. J. Stetter, "The defect correction principle and discretization methods," Numer. Math., v. 29, 1978, pp. 425-443. MR 0474803 (57:14436)
  • [13] K. Stüben & U. Trottenberg, Multigrid Methods: Fundamental Algorithms, Model Problem Analysis and Applications, in Lecture Notes in Math., vol. 960, Springer-Verlag, Berlin and New York, 1982, pp. 1-176. MR 685773 (84m:65129)

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Article copyright: © Copyright 1987 American Mathematical Society

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