Wavelet solutions for the Dirichlet problem
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 \textit{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!).
scientific article; zbMATH DE number 770165