## Wavelet bases adapted to pseudodifferential operators.(English)Zbl 0806.65112

The paper is concerned with the wavelet Galerkin method for the approximate solution of pseudodifferential equations in $$\mathbb{R}^ 2$$. Given a bilinear form $$a(u,v)$$ induced by a differential operator or a pseudodifferential operator and given a multiresolution analysis with generator $$\varphi$$, the authors construct a wavelet basis $$\{\psi_ e\}$$ such that $$(a(\varphi(\cdot- k), \psi_ e(\cdot- l))=0$$, $$k,l\in \mathbb{Z}^ 2$$, $$e\in E$$, where $$E$$ denotes the set of all vertices of the unit square different from (0,0). The first approach presented in the paper yields compactly supported wavelets under restrictive assumptions on the symbol of the operator. The second approach is based on the concept of biorthogonal wavelets and is more general.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35S15 Boundary value problems for PDEs with pseudodifferential operators
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