Adaptive wavelet schemes for nonlinear variational problems

SIAM Journal on Numerical Analysis

Volumn 41, Issue 5, 2003, Pages 1785-1823

  Cohen, Albert ^a   Dahmen, Wolfgang ^b   Devore, Ronald ^c  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^b RWTH AACHEN UNIVERSITY   (Germany)

^c UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Adaptive application of operators; Convergence rates; Gradient iteration; Mapping properties; Semilinear equations; Sparse evaluation of nonlinear mappings of wavelet expansions; Tree approximation; Variational problems; Wavelet representations

Indexed keywords

ADAPTIVE APPLICATION OF OPERATORS; CONVERGENCE RATES; GRADIENT ITERATION; MAPPING PROPERTIES; NEWTON'S SCHEME; SEMILINEAR EQUATIONS; SPARSE EVALUATION OF NONLINEAR MAPPINGS OF WAVELET EXPANSIONS; TREE APPROXIMATION; VARIATIONAL PROBLEMS; WAVELET REPRESENTATION;

ADAPTIVE SYSTEMS; APPROXIMATION THEORY; CONFORMAL MAPPING; CONVERGENCE OF NUMERICAL METHODS; NONLINEAR SYSTEMS; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

WAVELET TRANSFORMS;

EID: 9244259071     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902412269     Document Type: Conference Paper

References (24)

1. A. BARINKA, Fast Evaluation Tools for Adaptive Wavelet Schemes, Ph.D. Dissertation, Rheinisch Westfälische Technische Hochschule Aachen, Aachen, Germany, 2002, in preparation.
2. A. BARINKA, T. BARSCH, P. CHARTON, A. COHEN, S. DAHLKE, W. DAHMEN, AND K. UR-BAN, Adaptive wavelet schemes for elliptic problems-implementation and numerical experiments, SIAM J. Sci. Comput., 23 (2001), pp. 910-939.
3. P. BINEV AND R. DEVORE, Fast computation in adaptive tree approximation, Numer. Math., to appear.
4. P. BINEV, W. DAHMEN, AND R. DEVORE, Adaptive finite element methods with convergence rates, Numer. Math., to appear.
5. A. CANUTO, A. TABACCO, AND K. URBAN, The wavelet element method, part I: Construction and analysis, Appl. Comput. Harmon. Anal., 6 (1999), pp. 1-52.
6. A. CANUTO, A. TABACCO, AND K. URBAN, The wavelet element method, part II: Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal., 8 (2000), pp. 123-165.
7. A. COHEN, Numerical Analysis of Wavelet Methods, Studies in Mathematics and Its Applications 32, North-Holland, Amsterdam, 2003.
8. A. COHEN, W. DAHMEN, AND R. DEVORE, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp., 70 (2001), pp. 27-75.
9. A. COHEN, W. DAHMEN, AND R. DEVORE, Adaptive wavelet methods II: Beyond the elliptic case, Found. Comput. Math., 2 (2002), pp. 203-245.
10. A. COHEN, W. DAHMEN, AND R. DEVORE, Sparse evaluation of compositions of functions using multiscale expansions, SIAM J. Math. Anal., 35 (2003), pp. 279-303.
11. A. COHEN AND R. MASSON, Wavelet adaptive methods for second order elliptic problems, boundary conditions and domain decomposition, Numer. Math., 86 (2000), pp. 193-238.
12. S. DAHLKE, W. DAHMEN, R. HOCHMUTH, AND R. SCHNEIDER, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math., 23 (1997), pp. 21-47.
13. S. DAHLKE, W. DAHMEN, AND K. URBAN, Adaptive wavelet methods for saddle point problems: Optimal convergence rates, SIAM J. Numer. Anal., 40 (2002), pp. 1230-1262.
14. W. DAHMEN AND R. SCHNEIDER, Composite wavelet bases for operator equations, Math. Comp., 68 (1999), pp. 1533-1567.
15. W. DAHMEN AND R. SCHNEIDER, Wavelets on manifolds I: Construction and domain decomposition, SIAM J. Math. Anal., 31 (1999), pp. 184-230.
16. W. DAHMEN, R. SCHNEIDER, AND Y. Xu, Nonlinear functions of wavelet expansions: Adaptive reconstruction and fast evaluation, Numer. Math., 86 (2000), pp. 49-101.
17. W. DAHMEN, K. URBAN, AND J. VORLOEPER, Adaptive wavelet methods: Basic concepts and applications to the Stokes problem, in Wavelet Analysis, D.-X. Zhou, ed., World Scientific, River Edge, NJ, 2002, pp. 39-80.
18. P. DEUFLHARD AND M. WEISER, Local inexact Newton multilevel FEM for nonlinear elliptic problems, in Computational Science for the 21st Century (Tours, France), M.-O. Bristeau, G. Etgen, W. Fitzigibbon, J.-L. Lions, J. Periaux, and M. Wheeler, eds., Wiley-Interscience-Europe, London, 1997, pp. 129-138.
19. P. DEUFLHARD AND M. WEISER, Global inexact Newton multilevel FEM for nonlinear elliptic problems, in Multigrid Methods, Lecture Notes in Comput. Sci. Engrg. 3, W. Hackbusch and G. Wittum, eds., Springer-Verlag, New York, 1998, pp. 71-89.
20. R. DEVORE, Nonlinear approximation, in Acta Numerica, Acta Numer. 7, Cambridge University Press, Cambridge, UK, 1998, pp. 51-150.
21. J. POUSIN AND J. RAPPAZ, Consistency, stability, a-priori and a-posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math., 69 (1994), pp. 213-231.
22. M. RENARDY AND R.C. ROGERS, An Introduction to Partial Differential Equations, Texts Appl. Math. 13, Springer-Verlag, New York, 1993.
23. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.
24. E. ZEIDLER, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization, Springer-Verlag, New York, 1985.