Wavelet methods for fast resolution of elliptic problems.

*(English)*Zbl 0761.65083The 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.

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.

Reviewer: E.D’yakonov (Moskva)

##### 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 |