Multiscale methods for pseudodifferential equations

The present lecture which is essentially based on the papers [3]-[7] is concerned with generalized Petrov-Galerkin methods for elliptic periodic pseudodifferential equations in IR π covering classical Galerkin schemes, collocation, and others. These methods are based on a general setting of multiresolution analysis, i.e., of sequences of nested spaces which are generated by scaling functions (Sect. 2). In Sect. 3 we develop a general stability and convergence theory for such a framework which recovers and extends many previously studied special cases. The key to the analysis is a local principle due to one of the authors. Its applicability relies here on a sufficiently general version of a so called discrete commutator property of wavelet bases (see [3]). These results establish important prerequisites for developing and analysing methods for the fast solution of the resulting linear systems (Sect. 4). These methods are based on compressing the stiffness matrices relative to wavelet bases for the given multiresolution analysis. Such a compression technique has been proposed in [2] where, however, only classical Galerkin methods and operators of order zero were discussed. It is shown (see [4]) that the order of the overall computational work which is needed to realize a required accuracy is of the form o(N(logN) b ), where N is the number of unknowns and b ≥ 0 is some real number. In Sect. 5 the theoretical results are confirmed by new numerical experiments for the exterior Dirichlet problem for the Helmholtz equation.