Abstract
This paper develops a class of finite elements for compactly supported, shift-invariant functions that satisfy a dyadic refinement equation. Commonly referred to as wavelets, these basis functions have been shown to be remarkably well-suited for integral operator compression, but somewhat more difficult to employ for the representation of arbitrary boundary conditions in the solution of partial differential equations. The current paper extends recent results for treating periodized partial differential equations on unbounded domains in R n, and enables the solution of Neumann and Dirichlet variational boundary value problems on a class of bounded domains. Tensor product, wavelet-based finite elements are constructed. The construction of the wavelet-based finite elements is achieved by employing the solution of an algebraic eigenvalue problem derived from the dyadic refinement equation characterizing the wavelet, from normalization conditions arising from moment equations satisfied by the wavelet, and from dyadic refinement relations satisfied by the elemental domain. The resulting finite elements can be viewed as generalizations of the connection coefficients employed in the wavelet expansion of periodic differential operators. While the construction carried out in this paper considers only the orthonormal wavelet system derived by Daubechies, the technique is equally applicable for the generation of tensor product elements derived from Coifman wavelets, or any other orthonormal compactly supported wavelet system with polynomial reproducing properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alpert, B. K., 1992: Wavelets and other bases for fast numerical linear algebra. In: Chui, C. K. (ed): Wavelets: a tutorial in theory and applications, pp. 181–216: Academic Press
BankR. E.; DupontT. F.; YserentantH. 1988: The hierarchical basis multigrid method. Numer. Math. 52, 427–458
BeylkinG.; CoifmanR.; RokhlinV. 1991: Fast wavelet transforms and numerical algorithms I. Communications on Pure and Applied Mathematics. XLIV, 141–183
de Boor, Carl 1993: Multivariate piecewise polynomials. Acta Numerica, 65–109
Chui, C. K. 1992: Wavelets: A Tutorial in Theory and Applications: Academic Press.
DahlkeS.; KunothA. 1993: Biorthogonal Wavelets and Multigrid. Institut fur Geometrie and Praktische Mathematik, Bericht Nr. 84. Aachen, Germany
DahmenW.; ProssdorfS.; SchneiderR. 1992: Wavelet Approximation Methods for Pseudodifferential Equations I: Stability and Convergence. Institut fur Geometrie and Praktische Mathematik. Bericht Nr. 77. Aachen, Germany
DahmenW.; ProssdorfS.; SchneiderR. 1993a: Wavelet Approximation Methods for Pseudodifferential Equations II: Matrix Compression and Fast Solution. Institut fur Geometrie und Praktische Mathematik. Bericht Nr. 84. Aachen, Germany
DahmenW.; ProssdorfS.; SchneiderR. 1993b: Multiscale Methods for Pseudodifferential Equations. Institut fur Geometrie and Praktische Mathematik. Bericht Nr. 86. Aachen, Germany
DahmenW.; KunothA. 1992: Multilevel Preconditioning. Numer. Math. 63: 315–344
DahmenW.; and MicchelliC. A. 1993: Using the Refinement Equation for Evaluating Integrals of Wavelets. SIAM J. Numer. Anal. 30, 507–537
DaubechiesI. 1990: The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory. 36: 961–1005
DaubechiesI. 1992: Ten Lectures on Wavelets. Philadelphia: SIAM Publishing
DeVoreR.; JawerthB.; PopovV. 1992: Compression of wavelet decompositions. Amer. J. Math. 114: 737–785
DeVoreR.; JawerthB.; LucierB. 1992: Image Compression Through Wavelet Transform Coding. IEEE Transactions on Information Theory. 38: 719–746
Glowinski, R.; Lawton, W. M.; Ravachol, M.; Tenenbaum, E. 1989: Wavelet Solution of Linear and Nonlinear Elliptic, Parabolic and Hyperbolic Problems in One Space Dimension. Aware Inc. Technical Report AD890527.1
Glowinski, R.; Pan, T. W.; Wells, R. O.; Zhou, X. 1992: Wavelet and Finite Element Solutions for the Neumann Problem using Fictitious Domains. Computational Mathematics Laboratory, Rice University, Technical Report 92-01
HeurtauxF.; PlanchonF.; WickerhauserM. V. 1994: Scale Decomposition in Burger's Equation. In: BenedettoJ. J.; FrazierM. W. (ed): Wavelets: Mathematics and Applications, pp. 505–523. Boca Raton: CRC Press
Jaffard, S.; Laurencot, Ph. 1992: Orthonormal Wavelets, Analysis of Operators, and Applications to Numerical Analysis. In: Chui, C. (ed): Wavelets: A Tutorial in Theory and Applications, pp. 543–601: Academic Press
Jawerth, B. 1994: Wavelets on Closed Sets ..., preprint
Ko, J.; Kurdila, A. J. 1992: Connection Coefficient Truncation Error in Wavelet Differentiation. Center for Mechanics and Control, Department of Aerospace Engineering, Texas A & M University, Technical Report CMC-93-01
Ko, J.; Kurdila, A. J.; Park, S.; Strganac, T. W. 1993: Calculation of Numerical Boundary Measures for Wavelet Galerkin Approximations in Aeroelasticity. Proceedings of the 34th Structures, Structural Dynamics and Materials Conference
Ko, J.; Kim, C.; Kurdila A. J.; Strganac, T. W. 1993: Wavelet Galerkin Methods for Game Theoretic Control of Distributed Parameter Systems. Proceedings of the 34th Structures Structural Dynamics and Materials Conference
Ko, J.; Kurdila, A. J.; Pilant, M. S. 1994: A Class of Wavelet-based Finite Elements for Computational Mechanics. Proceedings of the 35th Structures, Structural Dynamics and Materials Conference
Ko, J.; Kurdila, A. J.; Wells, R. O.; Zhou, X. 1994: On the Stability of Numerical Boundary Measures in Wavelet Galerkin Methods. Proceedings of the 35th Structures, Structural Dynamics and Materials Conference
Kurdila, A. J. 1992: Symbolic Calculation of Wavelet Galerkin Quadratures. Center for Mechanics and Control, Department of Aerospace Engineering, Texas A & M University, Technical Report CMC TR92-01
Latto, A.; Resnikoff, H. L.; Tenenbaum, E. 1991: The Evaluation of Connection Coefficients of Compactly Supported Wavelets. Aware Inc., Technical Report AD910708
LeTallec, P.: Domain Decomposition Methods in Computational Mechanics. Computational Mechanics Advances, to appear: Elsevier
Park, S.; Kurdila, A. J. 1993: Wavelet Galerkin Multigrid Methods. Center for Mechanics and Control, Department of Aerospace Engineering, Texas A & M University, Technical Report CMC TR93-04
Resnikoff, H. L. 1991: Wavelets and Adaptive Signal Processing. Aware Inc., Technical Report AD910805
Rieder, A.; Wells, R. O.; Zhou, X. 1993: A Wavelet Approach to Robust Multilevel Solvers for Anisotropic Elliptic Problems. Computational Mathematics Laboratory, Rice University, Technical Report CML-93-07
Rieder, A. 1993: Semi-Algebraic Multilevel Methods Based Upon Wavelet Decompositions I: Application to Two-Point Boundary Value Problems. Computational Mathematics Laboratory, Rice University, Technical Report CML-93-04
Rieder, A.; Zhou, X. 1993: On the Robustness of the Damped V-Cycle of the Wavelet Frequency Decomposition Multigrid Method. Computational Mathematics Laboratory, Rice University, Technical Report CML-93-08
StrangG. 1989: Wavelets and Dilation Equations: A Brief Introduction. SIAM Review. 31: 614–627
Strang, G.; Fix, G. 1971: A Fourier Analysis of the Finite Element Variational Method. Constructive Aspects of Functional Analysis, pp. 793–840
Sweldens, W.; Piessens, R.: Asymptotic Error Expansion of Wavelet Approximations of Smooth Functions II: Generalization. preprint
SzaboB.; BabuskaI. 1991. Finite Element, Analysis. New York: John Wiley and Sons, Inc.
TraubJ. F.; WasilkowskiG. W.; WozniakowskiH. 1988: Information Based Complexity. Boston: Academic Press, Inc.
Wells, R. O.; Zhou, X. 1992a: Wavelet Solutions for the Dirichlet Problem. Computational Mathematics Laboratory, Rice University, Technical Report 92-02
Wells, R. O.; Zhou, X. 1992b: Wavelet Interpolation and Approximate Solutions of Elliptic Partial Differential Equations. Computational Mathematics Laboratory, Rice University, Technical Report 92-03
Wells, R. O.; Zhou, X. 1993: Representing the Geometry of Domains by Wavelets with Applications to Partial Differential Equations. Compuational Mathematics Laboratory, Rice University, Technical Report CML-92-14
YserentantH. 1986: On the Multi-Level Splitting of Finite Element Spaces. Numer. Math. 49: 379–412
YserentantH. 1990: Two Preconditioners Based on the Multilevel Splitting of Finite Element Spaces. Numer. Math. 58, 164–184
Author information
Authors and Affiliations
Additional information
Communicated by S. N. Atluri, 29 March 1995
Research supported in part by NASA Langley Research Center, Computational Structural Mechanics Branch, Jerry Housner
Rights and permissions
About this article
Cite this article
Ko, J., Kurdila, A.J. & Pilant, M.S. A class of finite element methods based on orthonormal, compactly supported wavelets. Computational Mechanics 16, 235–244 (1995). https://doi.org/10.1007/BF00369868
Issue Date:
DOI: https://doi.org/10.1007/BF00369868