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Wavelet solutions for the Dirichlet problem. (English) Zbl 0824.65108

The Dirichlet problem (for an elliptic equation, converted to the usual weak form) is solved by a penalty method, where the solution is sought in a slightly larger, simply shaped (rectangular in general) domain, using an \(L^ 2\)-extension of data in the equation. This formulation enables to work with a regular mesh grid and eliminates the need of generating a complex computation grid in the case of irregular domains.
The implementation is based on the use of compactly supported, differentiable wavelets introduced by I. Daubechies [Comm. Pure Appl. Math. 41, 901-966 (1988; Zbl 0644.42026)] generalizing the Haar basis. The main topic is an orthonormal representation of boundary data and of the geometry, i.e. of the characteristic function of the domain of the solution, by those wavelets. Convergence of all relevant approximations to the related data is proved, some simple examples (disc or diamond inside a square) are discussed, too.
(An unusually high part (appr. 60%) of the references cited here has not been published in the original resources of international discussion – books and journals –, but in lectures, tutorials, conference proceedings, or technical notes, or not appeared before the recent work - - no facility for the international scientific community!).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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