Abstract
The purpose of this paper is to investigate Galerkin schemes for the Stokes equations based on a suitably adapted multiresolution analysis. In particular, it will be shown that techniques developed in connection with shift-invariant refinable spaces give rise to trial spaces of any desired degree of accuracy satisfying the Ladyšenskaja-Babuška-Brezzi condition for any spatial dimension. Moreover, in the time dependent case efficient preconditioners for the Schur complements of the discrete systems of equations can be based on corresponding stable multiscale decompositions. The results are illustrated by some concrete examples of adapted wavelets and corresponding numerical experiments.
Zusammenfassung
In dieser Arbeit werden Galerkin-Verfahren für das Stokes-Problem untersucht, die auf speziell angepaßten Multiresolution-Ansätzen beruhen. Insbesondere wird gezeigt, daß gewisse Konstruktionsprinzipien für Wavelets auf gleichförmigen Gittern für jede Raumdimension und beliebige gewünschte Exaktheitsordnung auf Parre von Ansatzräumen führen, die die Ladyšenskaja-Babuška-Brezzi-Bedingung erfüllen. Darüber hinaus ergeben sich auch im instationären Fall aus den entsprechenden stabilen Multiskalenzerlegungen effiziente Vorkonditionierer für die Schurkomplemente entsprechenden systemmatrizen. Die Ergebnisse werden anhand einiger konkreter Realisierungen und numerischer Tests illustriert.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Andersson, L., Hall, N., Jawerth, B., Peters, G.: Wavelets on closed subsets of the real line. In: Topics in the theory and applications of wavelets (Schumaker, L. L., Webb, G. eds.), pp. 1–61. Boston: Academic Press, 1994.
de Boor, C.: A practical guide to splines. Berlin Heidelberg New York: Springer 1978.
Braess, D.: Finite Elemente. Berlin Heidelberg New York Tokyo: Springer 1992.
Bramble, J. H., Pasciak, J. E.: Iterative techniques for time dependent Stokes problems. Preprint, 1994.
Bramble, J. H., Pasciak, J. E., Xu, J.: Parallel multilevel preconditioners, Math. Comp.55, 1–22 (1990).
Cavaretta, A. S., Dahmen, W., Micchelli, C. A.: Stationary subdivision. Memoirs of the American Mathematical Society93, #453, 1991.
Chui, C. K., Quak, E.: Wavelets on a bounded interval. In: Numerical methods of approximation theory (Braess, D., Schumaker, L. L., eds.), pp. 1–24. Basel: Birkhäuser 1992.
Cohen, A., Dahmen, W., Devore, R. A.: Multiscale decompositions on bounded domains. IGPH-Preprint 113, RWTH Aachen, 1995.
Cohen, A., Daubechies, I.: A stability criterion for biorthogonal wavelet bases and their related subband coding scheme. Duke Math. J.68, 313–335 (1992).
Cohen, A., Daubechies, I., Jawerth, B., Vial, P.: Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris316, 417–421 (1993).
Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math.45, 485–560 (1992).
Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms, Appl. Comp. Horm. Anal.1, 54–81 (1993).
Dahmen, W.: Some remarks on multiscale transformations, stability and biorthogonality. In: Wavelets, images and surface fitting (Laurent, P. J., Le Méhauté, A., Schumaker, L. L., eds.), Wellesley: A K Peters 1994.
Dahmen, W.: Stability of multiscale transformations, RWTH Aachen, Preprint 1994. Fourier Anal. Appl. (to appear).
Dahmen, W.: Multiscale analysis, approximation and interpolation spaces. In: Approximation theory VIII (Chui, C. K., Schumaker, L. L., eds.), pp. 47–88. Singapore: World Scientific 1995.
Dahmen, W., Kunoth, A.: Multilevel preconditioning. Numer. Math.63, 315–344 (1992).
Dahmen, W., Kunoth, A., Urban, K.: Biorthogonal spline-wavelets on the interval — stability and moment conditions (in preparation).
Dahmen, W., Micchelli, C. A.: Using the refinement equation for evaluating integrals of wavelets, Siam J. Numer. Anal.30, 507–537 (1993).
Fortin, M.: An analysis of the convergence of mixed finite element methods, R.A.I.R.O. Anal. Numer.11, 341–354 (1977).
Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Berlin Heidelberg New York Tokyo: Springer 1986.
Jouini, A., Lemarié-Rieusset, P. G.: Analyses multirésolutions biorthogonales et applications. Ann. Inst. H., Poincaré Anal. Non Linéaire10, 453–476 (1993).
Kunoth, A.: Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers. Adv. Comput. Math.4, 145–170 (1995).
Kunoth, A.: Computing refinable integrals — Documentation of the program — Version 1.1, Technical Report ISC-95-02-Math, Texas A&M University, May 1995.
Lemarié-Rieusset, P. G.: Analyses multi-résolutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs à divergence nulle. Revista Mat. Iberoamericana8, 221–236 (1992).
Lemarié, P. G.: Fonctions a support compact dans les analyses multi-résolutions. Revista Mat. Iberoamericana7, 157–182 (1991).
Oswald, P.: On discrete norm estimates related to multilevel preconditioners in the finite element method. In: Constructive theory of functions, Proc. Int. Conf. Varna 1991 (Ivanov, K. G., Petrushev, P., Sendov, B., eds.), pp. 203–214. Sofia: Bulg. Acad. Sci. 1992.
Urban, K.: On divergence-free wavelets. Adv. Comput.4, 51–82 (1995).
Urban, K.: A wavelet-Galerkin algorithm for the driven-cavity-Stokes-problem in two space dimensions. In: Numerical modelling in continuum mechanics, Proc. Conf. Prague 1994 (Feistauer, M., Rannacher, R., Kozel, K., eds.), pp. 278–289. Prague: Charles University 1995.
Urban, K.: Multiskalenmethoden für das Stokes-Problem und angepaßte Wavelet-Basen. Thesis, RWTH Aachen, 1995.
Villemoes, L. F.: Sobolev regularity of wavelets and stability of iterated filter banks. In: Progress in wavelet analysis and applications, Proc. Int. Conf. Toulouse 1992 (Meyer, Y., Roques, S., eds.), pp. 243–251. Paris: Frontieres 1993.
Xu, J.: Theory of multilevel methods. Report AM 48, Department of Mathematics, Pennsylvania State University, 1989.
Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer., 285–326 (1993).
Author information
Authors and Affiliations
Additional information
The work of this author is supported by the Deutsche Forschungsgemeinschaft.
The work of this author is supported by the Graduiertenkolleg “Analyse und Konstruktion in der Mathematik” funded by the Deutsche Forschungsgemeinschaft at the RWTH Aachen.
Rights and permissions
About this article
Cite this article
Dahmen, W., Kunoth, A. & Urban, K. A wavelet Galerkin method for the stokes equations. Computing 56, 259–301 (1996). https://doi.org/10.1007/BF02238515
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02238515