SONDERFORSCHUNGSBEREICH 393: NUMERISCHE SIMULATION AUF MASSIV-PARALLELEN RECHNERN, TECHNISCHE UNIVERSITÄT CHEMNITZ

Beuchler, Sven ; Schneider, Reinhold ; Schwab, Christoph : Multiresolution weighted norm equivalences and applications

Author(s):

Beuchler, Sven
Schneider, Reinhold
Schwab, Christoph

Title:

Multiresolution weighted norm equivalences and applications

Electronic source:

application/pdf
application/postscript

Preprint series:

Technische Universität Chemnitz, SFB 393 (Germany), SFB393-Preprint 9, 2002

Mathematics Subject Classification:

65T60			[ Wavelets ]
65N30			[ Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods ]
65N22			[ Solution of discretized equations ]
65F35			[ Matrix norms, conditioning, scaling ]
65F50			[ Sparse matrices ]
65N35			[ Spectral, collocation and related methods ]

Abstract:

We establish multiresolution norm equivalences in weighted spaces $L^2_w((0,1))$ with possibly singular weight functions $w(x)\geq 0$ in $(0,1)$. Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function $w(x)$ within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning $p$-Version FEM and wavelet discretizations of degenerate elliptic problems.

Keywords:

wavelets, fast solvers, weighted norms

Language:

English

Publication time:

1 / 2003