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Adaptive wavelet methods for saddle point problems

Published online by Cambridge University Press:  15 April 2002

Stephan Dahlke ,
Reinhard Hochmuth  and
Karsten Urban
Stephan Dahlke
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. ([email protected])
Reinhard Hochmuth
Affiliation:
FU Berlin, FB Mathematik, Arnimallee 2-6, 14195 Berlin, Germany. ([email protected])
Karsten Urban
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. ([email protected])
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Abstract

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge.This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

Keywords

Adaptive schemesaposteriori error estimatesmultiscale methodswaveletssaddle point problemsUzawa's algorithm.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 5 , September 2000 , pp. 1003 - 1022
Copyright
© EDP Sciences, SMAI, 2000

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References

H.W. Alt, Lineare Funktionalanalysis (in german). Springer-Verlag, Berlin (1985).
K. Arrow, L. Hurwicz and H. Uzawa, Studies in Nonlinear Programming. Stanford University Press, Stanford, CA (1958).
Bertoluzza, S., A posteriori error estimates for the wavelet Galerkin method. Appl. Math. Lett. 8 (1995) 1-6. CrossRef
S. Bertoluzza and R. Masson, Espaces vitesses-pression d'ondelettes adaptives satisfaisant la condition Inf-Sup. C. R. Acad. Sci. Paris, Sér. Math. 323 (1996).
D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge University Press, Cambridge (1997).
Bramble, J.H., Pasciak, J.E. and Vassilev, A.T., Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34 (1997) 1072-1092. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
A. Cohen, Wavelet methods in Numerical Analysis, in: Handbook of Numerical Analysis, North Holland, Amsterdam (to appear).
A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet schemes for elliptic operator equations - Convergence rates, RWTH Aachen, IGPM Preprint 165, 1998. Math. Comput. (to appear).
Dahlke, S., Dahmen, W., Hochmuth, R. and Schneider, R., Stable multiscale bases and local error estimation for elliptic problems. Appl. Numer. Math. 23 (1997) 21-48. CrossRef
S. Dahlke, R. Hochmuth and K. Urban, Adaptive wavelet methods for saddle point problems, Preprint 1126, Istituto di Analisi Numerica del C. N. R. (1999).
S. Dahlke, R. Hochmuth and K. Urban, Convergent Adaptive Wavelet Methods for the Stokes Problem, in: Multigrid Methods VI, E. Dick, K. Riemslagh, J. Vierendeels Eds., Springer-Verlag (2000).
Dahmen, W., Stability of multiscale transformations. J. Fourier Anal. Appl. 2 (1996) 341-361.
Dahmen, W., Wavelet and multiscale methods for operator equations. Acta Numerica 6 (1997) 55-228. CrossRef
W. Dahmen, Wavelet methods for PDEs -- Some recent developments, RWTH Aachen, IGPM Preprint 183 (1999).
Dahmen, W., Kunoth, A. and Urban, K., Wavelet-Galerkin, A method for the Stokes problem. Computing 56 (1996) 259-302. CrossRef
Elman, H.C. and Golub, G.H., Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31 (1994) 1645-1661. CrossRef
Fortin, M., Old and new Finite Elements for incompressible flows. Int. J. Numer. Meth. Fluids 1 (1981) 347-364. CrossRef
Hochmuth, R., Stable multiscale discretizations for saddle point problems and preconditioning. Numer. Funct. Anal. and Optimiz. 19 (1998) 789-806. CrossRef
Lemarié-Rieusset, P.G., Analyses multi-résolutions non orthogonales, Commutation entre Projecteurs et Derivation et Ondelettes Vecteurs à divergence nulle. Rev. Mat. Iberoam. 8 (1992) 221-236. CrossRef
R. Masson, Wavelet discretizations of the Stokes problem in velocity-pressure variables, Preprint, Univ. P. et M. Curie, Paris (1998).
Urban, K., On divergence-free wavelets. Adv. Comput. Math. 4 (1995) 51-82. CrossRef
K. Urban, Wavelet bases in H(div) and H(curl) , Preprint 1106, Istituto di Analisi Numerica del C. N. R., 1998. Math. Comput. (to appear).