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Problem dependent generalized prewavelets. (English) Zbl 1239.65077
Summary: We present a new approach to construct robust multilevel algorithms for elliptic differential equations. The multilevel algorithms consist of multiplicative subspace corrections in spaces spanned by problem dependent generalized prewavelets. These generalized prewavelets are constructed by a local orthogonalization of hierarchical basis functions with respect to a so-called local coarse-grid space. Numerical results show that the local orthogonalization leads to a smaller constant in strengthened Cauchy-Schwarz inequality than the original hierarchical basis functions. This holds also for several equations with discontinuous coefficients. Thus, the corresponding multilevel algorithm is a fast and robust iterative solver.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
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