×

zbMATH — the first resource for mathematics

Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. (English) Zbl 0826.65099
The paper provides a thorough theoretical background for tensor-product multigrid methods. Application is made to the traditional nodal multigrid methods and to prewavelet methods, with emphasis on their applicability to anisotropic elliptic equations. Numerical examples are presented for several such schemes as applied to a strongly anisotropic reduced wave equation.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. Auscher, Wavelets with boundary conditions on the interval, in: [6], pp. 217–236. · Zbl 0794.65098
[2] O. Axelsson and V.A. Barker,Finite Element Solution of Boundary Value Problems. Theory and Computation (Academic Press, new York, 1984). · Zbl 0537.65072
[3] O.V. Besov, L.D. Kudryavzev, P.I. Lizorkin and S.M. Nikol’skij, Investigations in the theory of spaces of differentiable functions of several variables, Proc. Steklov Ins. Math. 1 (1990) 73–139.
[4] J.H. Bramble, J.E. Pasciak and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1–22. · Zbl 0703.65076 · doi:10.1090/S0025-5718-1990-1023042-6
[5] H.-J. Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, Dissertation, TU München (1992).
[6] C.K. Chui (ed.),Wavelets: A Tutorial in Theory and Applications (Academic Press, Boston, 1992). · Zbl 0744.00020
[7] C.K. Chui and J.Z. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992) 903–916. · Zbl 0759.41008 · doi:10.2307/2153941
[8] A. Cohen, I. Daubechies, B. Jawerth and P. Vial, Multiresolution analysis, wavelets and fast algorithms on the interval, C.R. Acad. Sci. Paris 316 (1993) 417–421. · Zbl 0768.42015
[9] S. Dahlke and A. Kunoth, A biorthogonal wavelet approach for solving boundary value problems, Preprint Nr. 85, Inst. f. Geom. Prakt. Math., RWTH Aachen (1993).
[10] S. Dahlke and A. Kunoth, Biorthogonal wavelets and multigrid, in:Adaptive Methods – Algorithms, Theory and Applications, Proc. 9th GAMM Seminar, eds. W. Hackbusch and G. Wittum, Notes on Numerical Fluid Mechanics 46 (Vieweg, Braunschweig, 1994) pp. 99–119. · Zbl 0809.65085
[11] S. Dahlke and I. Weinreich, Wavelet-Galerkin-methods: An adapted biorthogonal wavelet basis, Constr. Approx. 9 (1993) 237–262. · Zbl 0777.34015 · doi:10.1007/BF01198005
[12] S. Dahlke and I. Weinreich, Wavelet bases adapted to pseudodifferential operators, Appl. Comp. Harmonic Anal. 1 (1994) 267–283. · Zbl 0806.65112 · doi:10.1006/acha.1994.1014
[13] W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992) 315–344. · Zbl 0757.65031 · doi:10.1007/BF01385864
[14] W. Dahmen, S. Prössdorf and R. Schneider, Multiscale methods for pseudodifferential equations, in:Recent Advances in Wavelet Analysis, eds. L.L. Schumaker and G. Webb (Academic Press, New York, 1994) pp. 191–235. · Zbl 0788.65112
[15] M. Dryja, B. Smith and O. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, Report 638, Courant Inst., New York Univ. (1993), to appear in SIAM J. Numer. Anal. · Zbl 0818.65114
[16] G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns-Hopkins Univ. Press, Baltimore, 1983). · Zbl 0559.65011
[17] M. Griebel, A parallelizable and vectorizable multi-level algorithm on sparse grids, in:Parallel Algorithms for PDE, Proc. 6th GAMM Seminar, Kiel, ed. W. Hackbusch, Notes on Numerical Fluid Mechanics 31 (Vieweg, Braunschweig, 1991) pp. 94–100. · Zbl 0741.65090
[18] M. Griebel, Parallel multigrid methods on sparse grids, in:Multigrid Methods III, International Series of Numerical Mathematics 98 (Birkhäuser, Basel, 1991) pp. 211–221.
[19] M. Griebel, Multilevel algorithms considered as iterative methods on semidefinite systems, SIAM Int. J. Sci. Stat. Comp. 15 (1994) 547–565. · Zbl 0818.65108 · doi:10.1137/0915036
[20] M. Griebel,Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen, Teubner Skripten zur Numerik (Teubner, Stuttgart, 1994). · Zbl 0823.65026
[21] M. Griebel and P. Oswald, On additive Schwarz preconditioners for sparse grid discretizations, Numer. Math. 66 (1994) 449–464. · Zbl 0791.65019 · doi:10.1007/BF01385707
[22] M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., to appear. · Zbl 0826.65098
[23] M. Griebel and P. Oswald, Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems, SFB-report 342/15/94, Institut für Informatik, TU München (1994).
[24] M. Griebel, M. Schneider and C. Zenger, A combination technique for the solution of sparse grid problems, in:Iterative Methods in Linear Algebra, eds. P. de Groen and R. Beauwens, IMACS (Elsevier/North-Holland, 1992) pp. 263–281. · Zbl 0785.65101
[25] W. Hackbusch,Iterative Solution of Large Sparse Systems of Equations (Springer, New York, 1994). · Zbl 0789.65017
[26] W. Hackbusch, The frequency decomposition multi-grid method. Part I: application to anisotropic equations, Numer. Math. 56 (1989) 229–245. · Zbl 0673.65062 · doi:10.1007/BF01409786
[27] W. Hackbusch, The frequency decomposition multi-grid methods. Part II: Convergence analysis based on the additive Schwarz method, Numer. Math. 63 (1992) 433–453. · Zbl 0793.65089 · doi:10.1007/BF01385869
[28] S. Jaffard, Wavelet methods for fast resolution of elliptic equations, SIAM J. Numer. Anal. 29 (1992) 965–986. · Zbl 0761.65083 · doi:10.1137/0729059
[29] S. Jaffard and P. Laurencot, Orthogonal wavelets, analysis of operators, and applications to numerical analysis, in: [6], pp. 543–601. · Zbl 0764.65066
[30] B. Jawerth and W. Sweldens, Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations,Proc. 6th Copper Mountain Multigrid Conf. (April 1993).
[31] G.G. Lorentz and R.A. DeVore,Constructive Approximation, Grundlehren vol. 303 (Springer, Berlin, 1993). · Zbl 0797.41016
[32] P. Oswald,Multilevel Finite Element Approximation. Theory and Applications, Teubner Skripten zur Numerik (Teubner, Stuttgart, 1994). · Zbl 0830.65107
[33] P. Oswald, On estimates for one-dimensional spline approximation, in:Splines in Numerical Analysis, eds. J.W. Schmidt and H. Späth, Math. Research, vol. 52 (Akad.-Verlag, Berlin, 1989) pp. 111–124. · Zbl 0739.41015
[34] P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990) 43–64. · Zbl 0703.41018
[35] G. Plonka, Generalized spline wavelets, Preprint 94/8, FB Mathematik, Univ. Rostock (1994). · Zbl 0815.65012
[36] A. Rieder, R.O. Wells and X. Zhou, A wavelet approach to robust multilevel solvers for anisotropic elliptic problems, Preprint, Rice University, Houston (1993). · Zbl 0864.65073
[37] A. Rieder and X. Zhou, On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method, TR CML TRS 3-10, Rice University, Houston (1993).
[38] H.-J. Schmeisser, Vector-valued Sobolev and Besov spaces, in:Seminar Analysis of the Karl-Weierstraß-Institute 1985/86, eds. B.-W. -Schulze and H. Triebel, Teubner Texte Math. Bd. 96 (Teubner, Leipzig, 1987) pp. 4–44.
[39] H.-J. Schmeisser and H. Triebel,Topics in Fourier Analysis and Function Spaces (Akad. Verlagsges. Geest & Portig K.-G., Leipzig, 1987) ch. 2,4. · Zbl 0661.46024
[40] K. Stüben and U. Trottenberg, Multigrid methods: Fundamental algorithms, model problem analysis and applications, in:Multigrid Methods, Lecture Notes in Mathematics 960 (Springer, 1982). · Zbl 0562.65071
[41] V.N. Temlyakov,Approximation of Periodic Functions (Nova Science, New York, 1993). · Zbl 0899.41001
[42] C. Tong, T. Chan and C. Kuo, Multilevel filtering elliptic preconditioners, SIAM J. Sci. Stat. Comp. 13 (1992) 227–245. · Zbl 0745.65069 · doi:10.1137/0913012
[43] J. Wang, Convergence analysis of Schwarz algorithm and multilevel decomposition iterative methods II: non-selfadjoint and indefinite problems, SIAM J. Numer. Anal. 30 (1993) 953–970. · Zbl 0777.65066 · doi:10.1137/0730050
[44] J. Weidmann,Linear Operators in Hilbert Spaces (Springer, New York, 1980). · Zbl 0434.47001
[45] J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992) 581–613. · Zbl 0788.65037 · doi:10.1137/1034116
[46] H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numer. (1993) 285–326. · Zbl 0788.65108
[47] C. Zenger, Sparse grids, in:Parallel Algorithms for PDE, Proc. 6th GAMM Seminar, Kiel, ed. W. Hackbusch, Notes on Numerical Fluid Mechanics 31 (Vieweg, Braunschweig, 1991) pp. 241–251. · Zbl 0763.65091
[48] X. Zhang, Multilevel additive Schwarz methods, Numer. Math. 63 (1992) 521–539. · Zbl 0796.65129 · doi:10.1007/BF01385873
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.