An adaptive wavelet-vaguelette algorithm for the solution of PDEs
A fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function is presented. This algorithm is an ``adaptive inversion scheme''. It employs semi-implicit time discretization for a parabolic partial differential equation (PDE) and is direct. No linear systems need to be solved due to diagonalizing the stiffness matrix through an appropriate choice of test and trial functions in a Petrov-Galerkin method. The adaptive discretization exploits the compression property of wavelets. The new algorithm 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 and self-adaptive wavelet discretization in space. Results with full adaptivity of the spatial discretization are presented for a one-dimensional flame front as well as for a two-dimensional problem.
scientific article; zbMATH DE number 992132
scientific article