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Embedding and a priori wavelet-adaptivity forDirichlet problems

Published online by Cambridge University Press:  15 April 2002

Andreas Rieder
Andreas Rieder*
Affiliation:
Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM), Universität Karlsruhe, 76128 Karlsruhe, Germany. email: andreas.rieder@math.uni-karlsruhe.de
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Abstract

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar, but rather theoretical, concepts already described in the literature our approach combines a high generality with an easy-to-implement algorithm.

Keywords

Boundary value problemfictitious domainGalerkin schemebiorthogonal wavelet systemadapted grid.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1189 - 1202
Copyright
© EDP Sciences, SMAI, 2000

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References

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. Tech. Report 173, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 52056 Aachen, Germany (1999).
Beylkin, G., Coifman, R. and Rokhlin, V., Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44 (1991) 141-183. CrossRef
Bramble, J.H. and Hilbert, S.R., Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112-124. CrossRef
Cai, W. and Zhang, W., An adaptive spline wavelet ADI(SW-ADI) method for two-dimensional reaction diffusion equations. J. Comput. Phys. 139 (1998) 92-126. CrossRef
Canuto, C., Tabacco, A. and Urban, K., The wavelet element method, part I: construction and analysis. Appl. Comput. Harmon. Anal. 6 (1999) 1-52. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Stud. Math. Appl. 4, North-Holland, Amsterdam (1978).
A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations - convergence rates. Math. Comp. posted on May 23, 2000, PII S0025-5718(00)01252-7 (to appear in print).
Cohen, A., Daubechies, I. and Feauveau, J.-C., Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992) 485-560. CrossRef
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, V. Latour and K. Gröchenig, Biorthogonal box spline wavelet bases, in Surface Fitting and Multiresolution Methods, A.L. Méhauté, C. Rabut and L.L. Schumaker Eds., Vanderbilt University Press (1997) 83-92.
Dahmen, W., Stability of multiscale transformations. J. Fourier Anal. Appl. 2 (1996) 341-362.
Dahmen, W., Wavelet and multiscale methods for operator equations. Acta Numer. 6 (1997) 55-228. CrossRef
W. Dahmen, A. Kurdila and P. Oswald Eds., Multiscale Wavelet Methods for Partial Differential Equations. Wavelet Anal. Appl. 6, Academic Press, San Diego (1997).
Dahmen, W., Prössdorf, S. and Schneider, R., Wavelet approximation methods for pseudodifferential equations. II. Matrix compression and fast solution. Adv. Comput. Math. 1 (1993) 259-335. CrossRef
Dahmen, W. and Schneider, R., Composite wavelet bases for operator equations. Math. Comp. 68 (1999) 1533-1567. CrossRef
Dahmen, W. and Stevenson, R., Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. 37 (1999) 319-352. CrossRef
Daubechies, I., Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 906-966.
I. Daubechies, Ten Lectures on Wavelets. CBMS-NSF Ser. in Appl. Math. 61, SIAM Publications, Philadelphia (1992).
Fröhlich, J. and Schneider, K., An adaptive wavelet-Galerkin algorithm for one- and two-dimensional flame computations. Eur. J. Mech. B Fluids 11 (1994) 439-471.
Fröhlich, J. and Schneider, K., An adaptive wavelet-vaguelette algorithm for the solution of nonlinear PDEs. J. Comput. Phys. 130 (1997) 174-190. CrossRef
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Ser. Comput. Phys., Springer-Verlag, New York (1984).
R. Glowinski, Finite element methods for the numerical simulation of incompressible viscous flow: Introduction to the control of the Navier-Stokes equations, in Vortex Dynamics and Vortex Methods, C.R. Anderson and C. Greengard Eds., Lectures in Appl. Math. 28, Providence, AMS (1991) 219-301.
Glowinski, R., Pan, T.-W. and Périaux, J., A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283-303. CrossRef
Glowinski, R., Pan, T.-W. and Périaux, J., Lagrange, A multiplier/fictitious domain method for the Dirichlet problem - generalizations to some flow problems. Japan J. Indust. Appl. Math. 12 (1995) 87-108. CrossRef
R. Glowinski, T.-W. Pan and J. Périaux, Fictitious domain methods for the simulation of Stokes flow past a moving disk, in Computational Fluid Dynamics '96, J.A. Desideri, C. Hirsh, P. LeTallec, M. Pandolfi and J. Périaux Eds., Chichester, Wiley (1996) 64-70.
W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment. Springer Ser. Comput. Math. 18, Springer-Verlag, Heidelberg (1992).
Jaffard, S., Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29 (1992) 965-986. CrossRef
Jaffard, S. and Meyer, Y., Bases d'ondelettes dans des ouverts de $\mathbb{R}^n$ . J. Math. Pures Appl. 68 (1992) 95-108.
A.K. Louis, P. Maass and A. Rieder, Wavelets: Theory and Applications. Pure Appl. Math., Wiley, Chichester (1997).
Y. Meyer, Ondelettes et Opérateurs I: Ondelettes. Actualités Mathématiques, Hermann, Paris (1990). English version: Wavelets and Operators, Cambridge University Press (1992).
P. Oswald, Multilevel solvers for elliptic problems on domains, in Dahmen et al. [] 3-58.
A. Rieder, On embedding techniques for 2nd-order elliptic problems, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler Eds., Wiley, Chichester (1997) 179-188.
Rieder, A., A domain embedding method for Dirichlet problems in arbitrary space dimension. RAIRO Modél. Math. Anal. Numér. 32 (1998) 405-431. CrossRef
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser. 22, Princeton University Press, Princeton (1970).
J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge, UK (1987).