Cohen, A.; Dahmen, W.; DeVore, R. Adaptive wavelet methods. II: Beyond the elliptic case. (English) Zbl 1025.65056 Found. Comput. Math. 2, No. 3, 203-245 (2002). [For part I see Math. Comput. 70, 27–75 (2001; Zbl 0980.65130)].Design and analysis of adaptive wavelet methods for systems of operator equations is presented. The existing theory on adaptive wavelet-based methods is extended from symmetric positive definite problems to indefinite or nonsymmetric systems of operator equations. This is accomplished by introducing techniques (such as the least-squares formulation) that transform the original (continuous problem) into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. For a wide range of problems it is shown that the new adaptive method performs with asymptotically optimal complexity. Furthermore it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of spaces, like the Ladyshenskaya-Babuška-Brezzi condition, no longer arise. Reviewer: Wilhelm Heinrichs (Essen) Cited in 6 ReviewsCited in 99 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65T60 Numerical methods for wavelets 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65Y20 Complexity and performance of numerical algorithms 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) Keywords:wavelets; adaptive methods; indefinite problems; convergence; quasi-sparse matrices; stability; operator equations; complexity; Ladyshenkaya-Babuška-Brezzi condition Citations:Zbl 0980.65130 PDFBibTeX XMLCite \textit{A. Cohen} et al., Found. Comput. Math. 2, No. 3, 203--245 (2002; Zbl 1025.65056) Full Text: DOI