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Adaptive wavelet algorithms for elliptic PDE’s on product domains. (English) Zbl 1127.41009
Using the methods of wavelets, tensor products and matrix compression, efficient methods for the approximation of solutions of elliptic partial differential equations are designed in this article. The key ingredients are tensor product, piecewise differentiable wavelets in higher dimensions and matrix compression. The wavelets stem from spaces of piecewise polynomials (splines). Accordingly, the problem is studied specifically on product domains (tensor domains) in \(d\)-dimensional real space. Several applications of this highly useful, adaptive wavelet ansatz are also studied in this paper, as well as the approaches to the quadrature required and the computability of the method.

MSC:
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
65T60 Numerical methods for wavelets
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
65D32 Numerical quadrature and cubature formulas
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
41A25 Rate of convergence, degree of approximation
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