Composite Wavelet Bases with Extended Stability and Cancellation Properties

The efficient solution of operator equations using wavelets requires that they generate a Riesz basis for the underlying Sobolev space and that they have cancellation properties of a sufficiently high order. Suitable biorthogonal wavelets were constructed on reference domains as the $n$-cube. Via a domain decomposition approach, these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds, where, in order to end up with local wavelets, biorthogonality was realized with respect to a modified $L_2$-scalar product. The use of this modified scalar product restricts the application of these so-called composite wavelets to problems of orders strictly larger than $-1$. Moreover, those wavelets with supports that extend to more than one patch generally have no cancellation properties. In this paper, we construct local, composite wavelets that are close to being biorthogonal with respect to the standard $L_2$-scalar product. As a consequence, they generate Riesz bases for the Sobolev spaces $H^s$ for the full range of $s$ allowed by the continuous gluing of functions over the patch interfaces, the properties of the primal and dual approximation spaces on the reference domain, and, in the manifold case, by the regularity of the manifold. Moreover, all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain. We illustrate our findings by a concrete realization of wavelets on a perturbed sphere.

[1]  Wolfgang Dahmen,et al.  Composite wavelet bases for operator equations , 1999, Math. Comput..

[2]  W. Dahmen.  Stability of Multiscale Transformations. , 1995 .

[3]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[4]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[5]  Rob Stevenson,et al.  On the Compressibility of Operators in Wavelet Coordinates , 2004, SIAM J. Math. Anal..

[6]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Optimal Complexity Estimates , 2006 .

[7]  Rob Stevenson,et al.  Locally Supported, Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes , 2000 .

[8]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[9]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[10]  Rob Stevenson,et al.  Finite‐element wavelets on manifolds , 2003 .

[11]  Albert Cohen,et al.  Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition , 2000, Numerische Mathematik.

[12]  Rob Stevenson,et al.  Stable three-point wavelet bases on general meshes , 1998, Numerische Mathematik.

[13]  Angela Kunoth,et al.  Wavelets on manifolds: An optimized construction , 2006, Math. Comput..

[14]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[15]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[16]  Wolfgang Dahmen,et al.  Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions , 1999, SIAM J. Numer. Anal..

[17]  Wolfgang Dahmen,et al.  Local Decomposition of Refinable Spaces and Wavelets , 1996 .

[18]  Hoang-Ngan Nguyen,et al.  Finite element wavelets for solving partial differential equations , 2005 .

[19]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[20]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .

[21]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[22]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[23]  Rob P. Stevenson,et al.  An optimal adaptive wavelet method without coarsening of the iterands , 2006, Math. Comput..

[24]  Panayot S. Vassilevski,et al.  Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results , 1998, SIAM J. Sci. Comput..