Adaptive wavelet methods for elliptic operator equations with nonlinear terms

Advances in Computational Mathematics

Volumn 19, Issue 1-3, 2003, Pages 99-146

  Xu, Yuesheng ^a,b   Zou, Qingsong ^a  

^a WEST VIRGINIA UNIVERSITY   (United States)

^b CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Adaptive wavelet methods; Constructive greedy schemes; Tree approximation

Indexed keywords

EID: 0037240053     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022816511874     Document Type: Conference Paper

References (39)

1. I. Babuska and A. Miller, A feedback finite element method with a-posterior error estimation: Part I, the finite element method and some basic properties of the a-posterior error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987) 1-40.
2. I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736-754.
3. A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet schemes for elliptic problems-implementation and numerical experiments, SIAM J. Sci. Comput. 23 (2001) 910-939.
4. J. Bergh and J. Lofstrom, Interpolation Spaces (Spring, Berlin, 1976).
5. S. Bertoluzza, A posterior error estimates for wavelet Galerkin methods, Appl. Math. Lett. 8 (1995) 1-6.
6. G. Beylkin, R.R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991) 141-183.
7. α, Mat. Sb. 73 (1967) 295-317.
8. F. Bomemann, B. Erdmann and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal. 33 (1996) 1188-1204.
9. Z. Chen, C. Micchelli and Y. Xu, A multilevel method for solving operator equations, J. Math. Anal. Appl. 262 (2001) 688-699.
10. Z. Chen, C. Micchelli and Y. Xu, Fast collocation methods for second kind integral equations, SIAM J. Numer. Anal. 40 (2002) 344-375.
11. A. Cohen, W. Dahmen, I. Daubechies and R. DeVore, Tree approximation and optimal encoding, Appl. Comput. Harmon. Anal. 11 (2001) 192-226.
12. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp. 70 (2001) 27-75.
13. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet schemes for nonlinear variational problems, IGPM Report, RWTH Aachen (June 2002).
14. A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity, SIAM J. Sci. Comput. 21 (1999) 1006-1026.
15. J. Conway, A Course in Functional Analysis, 2nd ed. (Springer, New York, 1990).
16. S. Dahlke, W. Dahmen and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Wavelet Multiscale Methods for PDEs (Academic Press, London, 1997) pp. 237-283.
17. S. Dahlke, W. Dahmen, R. Hochmuch and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23 (1997) 21-47.
18. S. Dahlke and R. DeVore, Besov regularity for elliptic boundary value probelems, Comm. Partial Differential Equations 22 (1997) 1-16.
19. W. Dahmen, Adaptive approximation by multivariate smooth splines, J. Approx. Theory 36 (1982) 119-140.
20. W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica 6 (1997) 55-228.
21. W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 86 (2000) 49-101.
22. W. Dahmen and C.A. Micchelli, Using the refinement equation for evaluating integral of wavelets, SIAM J. Numer. Anal. 30 (1993) 507-537.
23. W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993) 259-335.
24. W. Dahmen, S. Prösdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994) 583-620.
25. W. Dahmen, R. Schneider and Y. Xu, Nonlinear functionals of wavelet expansions-adaptive reconstruction and fast evaluation, Numer. Math. 86 (2000) 49-101.
26. G. Davis, S. Mallat and M. Avellaneda, Adaptive greedy approximations, Constr. Approx. 13 (1997) 57-98.
27. R. DeVore, Nonlinear approximation, Acta Numerica 7 (1998) 51-150.
28. R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737-785.
29. R. DeVore and G.G. Lorentz, Constructive Approximation (Springer, New York, 1991).
30. R. DeVore and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1998) 397-414.
31. R. DeVore and V. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math. 5 (1996) 173-187.
32. R. DeVore and X. Yu, Degree of adaptive approximation, Math. Comp. 55 (1990) 625-635.
33. R. Gribonval and M. Nielsen, Approximate weak greedy algorithms, Adv. Comput. Math. 14 (2001) 361-378.
34. H. Johen and K. Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, in: Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, Vol. 571 (Springer, New York, 1977) pp. 119-140.
35. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations (de Gryuter, Berlin, 1996).
36. V. Temlyakov, Greedy algorithms in Banach space, Adv. Comput. Math. 14 (2001) 277-292.
37. J. Xu and A. Zhou, Local and parallel finite element algorithms; based on two-grid discretizations for nonlinear problems, Adv. Comput. Math. 14 (2001) 293-327.
38. Y. Xu and Q. Zou, Greedy algorithms for construction of tree approximation, in preparation.
39. E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications (Springer, New York, 1995).