Adaptive Wavelet Collocation for Nonlinear BVPs
S. Bertoluzza, P. Pietra
One of the difficulties that one encounters in the application of wavelet based adaptive schemes to the numerical solution of partial differential equations is evaluating the action of nonlinear terms.
If the nonlinearity is of multilinear type (as u du/dx in Burgers equation) then one can apply the results by Dahmen and Micchelli, which allow to take such terms into account in a Galerkin scheme. When the nonlinearity is more complex, one has to pass through the physical space, by means of some approximate procedure consisting in evaluating the integrals of nonlinear expressions through some quadrature formula. In both approaches some fundamental quantities are computed on an uniform fine discretization, and the efficiency of the scheme is therefore reduced.
In this paper we propose a way to overcome such a difficulty. It is based on two different ingredients: first, a collocation scheme, which makes use of the nodal
values at collocation points and allows to avoid the computation of
integrals, and, second, the choice of a so-called {\sl interpolating wavelet basis}, for which the degrees of freedom are the nodal values themselves.
Both approaches allow us to work on the adapted space at all stages of the procedure.
We test the proposed method on a one-dimensional Euler-Poisson system for
semiconductor devices.
Proc. of ICAOS '96, Lecture Notes in Control and Information Sciences, Springer Verlag, London (1996) 13 n.1 (1994)
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