Wavelet Galerkin multigrid methods
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This paper derives multilevel preconditioners for a class of wavelet-based finite elements. The derivation extends recent multigrid approaches for periodic wavelet Galerkin methods to incorporate a wider class of boundary conditions. The derived multilevel preconditioners are similar in form to those used in periodic multigrid methods in that the restriction operators among levels are expressed precisely in terms of the filter coefficients commonly employed in image compression via multiresolution analysis. In contrast, however, the restriction and prolongation pair are not orthogonal over the domain of interest. Still, using recent discrete norm characterizations of Sobolev-Besov spaces in terms of wavelet expansion coefficients, it is possible to derive an additive Schwarz multilevel method whose convergence rate is asymptotically independent of discretization level.