This paper is concerned with the development of a self-adaptive spatial descretization for PDEs using a wavelet basis. A Petrov-Galerkin method [LPT91] is used to reduce the determination of the unknown at the new time step to the computation of scalar products. These have to be discretized in an appropriate way. We investigate this point in detail and devise an algorithm that has linear operation count with respect to the number of unknowns. It is tested with spline wavelets and Meyer wavelets retaining the latter for their better localisation at finite precision. The algorithm is then applied to the one dimensional thermodiffusive equations. We show that the adaption strategy merits to be modified in order to take into account the particular and very strong nonlinearity of this problem. Finally, a supplementary Fourier discretization permits the computation of two dimensional flame fronts.