Collection of abstracts 14th GAMM-Seminar Kiel on Concepts of Numerical Software January 23rd to 25th, 1998.

Friday, January 23rd, 1998

Software tools for using wavelets on the interval for the numerical solution of operator equations

Multiscale methods and wavelets offer some promising features for the numerical solution of certain operator equations such as integral equations and elliptic partial differential equations. For a long time, these applications were restricted to simple domains (such as the torus) since there was no construction of wavelet bases on general domains that could efficiently be used in implementations. Quite recently, several papers appeared using a domain decomposition technique by splitting the domain of interest into subdomains that are images of a single reference square or cube under certain parametric mappings, [CTU1,CTU2,DS1,DS2].
On the reference domain, one uses tensor products of wavelet bases on the interval as in [DKU] with appropriate boundary conditions.

Hence, the usefulness of such methods strongly depends on efficient software for dealing with wavelet systems on the interval. Since the realization of wavelet methods essentially differs from using finite element meshes, it is neccessary to develop new concepts for their implementation.
In this talk, we describe the main ingredients of our software and present an example of its usage.

[CTU1]  C. Canuto, A. Tabacco, and K. Urban}, 
        The Wavelet Element
        Method, Part I: Construction and analysis, 
        Istituto di Analisi Numerica del CNR, Pavia,
        Preprint No. 1038, 1997.

[CTU2]  C. Canuto, A. Tabacco, and K. Urban, 
        he Wavelet Element
        Method, Part II: Realization and additional features
        in 2D and 3D, 
        Istituto di Analisi Numerica del CNR, Pavia,
        Preprint No. 1052, 1997.

[DKU]   W. Dahmen, A. Kunoth, and K. Urban,
        Biorthogonal spline--wavelets on the interval ---
        Stability and moment conditions,
        RWTH Aachen, IGPM Preprint No. 129, 1996,  
        Appl. Comp. Harm. Anal., to appear.

[DS1]   W. Dahmen and R. Schneider, 
        Composite wavelet bases for operator equations, 
        RWTH Aachen, IGPM Preprint No. 133, 1996.

[DS2]   W. Dahmen and R. Schneider, 
        Multiscale methods
        for boundary integral equations I: Biorthogonal wavelets on
        2D--manifolds in $\er^3$, 
        in preparation.