An adaptive multilevel wavelet collocation method for elliptic problems

Abstract

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 OðNÞ, 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.

An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

“…This would allow us to solve the full incompressible NavierStokes equations on the sphere, taking advantage of wavelet multilevel decomposition and compression. In flat geometries this has been done successfully in [14]. This work is currently underway.…”

“…It would be interesting to combine the level set idea with the well-established adaptive wavelet collocation method on a three-dimensional Cartesian grid [14].…”

“…In flat geometries the adaptive wavelet collocation method (AWCM) was pioneered by Bertoluzza [11] for time-independent problems, and has since been developed for more general problems using second generation wavelets [12][13][14][15]. This approach combines the adaptivity and error control of the adaptive wavelet method with the flexibility of collocation.…”

On interpolatory divergence-free wavelets

“…The adaptation and usage of second-generation wavelets for the numerical solution of partial differential equations by a collocation method is introduced in [22,23,36,37].…”

“…This is one reason for considering collocation methods; see e.g. [22,23,35,37]. In this case, interpolatory wavelets are required, which can easily be adapted to complicated domains.…”