An adaptive multilevel wavelet collocation method for elliptic problems

An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is O ( N ) , where N is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two- and three-dimensional elliptic test problems.