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Complexity of an algorithm for solving saddle-point systems with singular blocks arising in wavelet-Galerkin discretizations. (English) Zbl 1099.65150
Summary: The paper deals with fast solving of large saddle-point systems arising in wavelet-Galerkin discretizations of separable elliptic PDEs. The periodized orthonormal compactly supported wavelets of the tensor product type together with the fictitious domain method are used. A special structure of matrices makes it possible to utilize the fast Fourier transform that determines the complexity of the algorithm. Numerical experiments confirm theoretical results.

65T60 Numerical methods for wavelets
65F10 Iterative numerical methods for linear systems
65T50 Numerical methods for discrete and fast Fourier transforms
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991. · Zbl 0788.73002
[2] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.
[3] M. Fortin, R. Glowinski: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam, 1983. · Zbl 0525.65045
[4] I. Daubechies: Ten Lectures on Wavelets. SIAM, Philadelphia, 1992. · Zbl 0776.42018
[5] R. Glowinski, T. Pan, and J. Periaux: A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111 (1994), 283–303. · Zbl 0845.73078
[6] R. Glowinski, T. Pan, R. O. Wells, X. Zhou: Wavelets methods in computational fluid dynamics. In: Proc. Algorithms Trends in Computational Dynamics (1993) (M. Y. Hussaini, A. Kumar, and M. D. Salas, eds.). Springer-Verlag, New York, pp. 259–276.
[7] G. H. Golub, C. F. Van Loan: Matrix Computation. The Johns Hopkins University Press, Baltimore, 1996, 3rd ed.
[8] N. I. M. Gould: On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem. Math. Program. 32 (1985), 90–99. · Zbl 0591.90068
[9] R. Kucera: Wavelet solution of elliptic PDEs. In: Proc. Matematyka v Naukach Technicznych i Przyrodniczych (2000) (S. Bialas, ed.). AGH Krakow, pp. 55–62.
[10] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York, 1987, 3rd ed. · Zbl 0925.00005
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