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Wavelet methods for fast resolution of elliptic problems. (English) Zbl 0761.65083
The main results of the paper are connected with an analysis of the condition number $$\kappa$$ of the Gram matrix $${\mathcal M}={\mathcal M}^*>0$$ associated with the energetic inner product for a second-order elliptic operator and a special choice of the basis of the approximating subspace.
The so called hierarchical bases are very popular nowadays. The author investigates another procedure leading to an $${\mathcal L}^ 2$$-orthonormal basis constructed from the standard finite-element basis composed of piecewise polynomials of degree $$2m-1$$ in each variable at cubical cells.
The basic theorem states that $$\kappa$$ can be estimated uniformly with respect to the grids. It enables one to use diagonal preconditioners. Local refinement of the grid is permitted. Some decay estimates for the entries of $$\mathcal M$$ are given as a compensation for the lack of sparsity of $$\mathcal M$$. Numerical examples deal with one space variable.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling 35J25 Boundary value problems for second-order elliptic equations
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