A Dynamical
Adaptive Concept Based on Wavelet Packet Best Bases:
Application to the Resolution
of Convection-Diffusion PDEs
Pascal Joly, Yvon
Maday and Valérie Perrier
We exploit in this paper a methodology based on the wavelet packet
concept. It allows for solving with very few number of degrees of freedom
partial differential equations. The main application is here the Burgers
equation with a small viscosity. The wavelet packet framework allows to
define the notion of a minimal basis that has proven to be an efficient
procedure for data compression. The purpose here is to take benefit of
this compression to represent accurately and economically the solution
of a time dependant PDE. The time discretization is a standard multistep
scheme. The spacial discretization is defined by infering a reduced basis
for the solution at the new time step, from the knowledge of the previous
ones. The wavelet packet method is a better approach for adaptivity in
the case where the solution to be approximated has many singularities.
"Multiscale wavelet methods for PDEs", W.
Dahmen, A. Kurdilla and P. Oswald (eds), Academic Press,
pp 199-235 (1997).
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