IJPAM: Volume 44, No. 1 (2008)

ADAPTIVE SOLUTION OF MULTIDIMENSIONAL PDES
VIA TENSOR PRODUCT WAVELET DECOMPOSITION

A. Averbuch$^1$, G. Beylkin$^2$, R. Coifman$^3$, P. Fischer$^4$, M. Israeli$^5$
$^1$School of Mathematical Sciences
Tel Aviv University
Tel Aviv, 69978, ISRAEL
$^2$Program in Applied Mathematics
University of Colorado at Boulder
Boulder, CO 80309-0526, USA
$^3$Department of Mathematics
Yale University
P.O. Box 2155, Yale Station, New Haven, CT 06520, USA
$^4$School of Mathematics and Computer Science
University Bordeaux 1
351 Cours de la Libération, Talence Cedex, 33405, FRANCE
$^5$Faculty of Computer Science
Technion - Israel Institute of Technology
Haifa, 32000, ISRAEL


Abstract.An adaptive multidimensional algorithm for solving strictly elliptic PDEs is described. The main points of the present approach are: 1) For a given finite (but arbitrary) accuracy $\epsilon$ we find scales and locations of $\epsilon$-significant wavelet coefficients by examining the wavelet decompositions of the right-hand side as well as the coefficients of the equation. In doing so we effectively construct subspaces for the solution (or ``masks") for different accuracy thresholds. 2) We use a diagonally preconditioned ``constrained" conjugate gradient (CG) method where the preconditioner is that of an extended problem with periodic bondary conditions to compute the actual coefficients of the approximate solution. We use sparse data structures in our implementations in order to take advantage of $O(N_s)$ complexity of the algorithm, where $N_s$ is the number of significant coefficients required for a given accuracy. We present numerical examples in one, two and three spatial dimensions.

Professor Moshe Israeli passed away on February 18, 2007.
This paper is dedicated to his memory.


Received: January 26, 2008

AMS Subject Classification: 35D05

Key Words and Phrases: multidimensional wavelet decomposition, adaptive discretization, preconditioned constrained conjugate gradient, sparse implementation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 44
Issue: 1