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Wavelets and the numerical solution of boundary value problems. (English) Zbl 0769.65083

Summary: We present a new numerical method for the solution of partial differential equations in nonseparable domains. The method uses a wavelet-Galerkin solver with a nontrivial adaptation of the standard capacitance matrix method. The numerical solutions exhibit spectral convergence with regard to the order of the compactly-supported, Daubechies wavelet basis. Furthermore, the rate of convergence is found to be independent of the geometry. We solve the Helmholtz equation since, for the indefinite case, the solutions have qualitative properties that well illustrate the applications of our method.

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
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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