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   -> Volume 4, Issue 12

Preprint: Adaptive Wavelet-Vaguelette Algorithm for PDEs
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froehlich@ZIB-Berlin.DE (Jochen Froehlich)

PostPosted: Fri Dec 06, 2002 9:37 am    
Subject: Preprint: Adaptive Wavelet-Vaguelette Algorithm for PDEs
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Preprint: Adaptive Wavelet-Vaguelette Algorithm for PDEs

Title: An adaptive Wavelet--Vaguelette Algorithm for the
Solution of Nonlinear PDEs

Authors: Jochen Fr"ohlich, Kai Schneider

Abstract: The paper describes a fast algorithm for the discrete
periodic wavelet transform and its inverse without using the scaling
function. The approach permits to compute the decomposition of a
function into a lacunary wavelet basis, i.e. a basis constituted of a
subset of all basis functions up to a certain scale, without
modification. The construction is then extended to operator--adapted
biorthogonal wavelets. This is relevant for the solution of
non--linear evolutionary PDEs where a priori information about the
significant coefficients is available. We pursue the approach
described in [FrSc94] which is based on the explicit computation of
the scalewise contributions of the approximated function to the values
at points of hierarchical grids. Here, we present an improved
construction employing the cardinal function of the
multiresolution. The new method is applied to the Helmholtz equation
and illustrated by comparative numerical results. It is then extended
for the solution of a nonlinear parabolic PDE with semi--implicit
discretization in time and self--adaptive wavelet discretization in
space. Results with full adaptivity of the spatial wavelet
discretization are presented for a one--dimensional flame front as
well as for a two--dimensional problem.

Konrad-Zuse-Zentrum Berlin, Preprint SC 95-28

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