MULTISCALE METHODS FOR PSEUDODIFFERENTIAL-EQUATIONS

The present lecture which is essentially based on the papers [3]-[7] i s concerned with generalized Petrov-Galerkin methods for elliptic peri odic pseudodifferential equations in IR(n) covering classical Galerkin schemes, collocation, and others. These methods are based on a genera l setting of multiresolution analysis, i.e., of sequences of nested sp aces which are generated by scaling functions (Sect. 2). In Sect. 3 we develop a general stability and convergence theory for such a framewo rk which recovers and extends many previously studied special cases. T he key to the analysis is a local principle due to one of the authors. As applicability relies here on a sufficiently general version of a s o called discrete commutator property of wavelet bases (see [3]). Thes e results establish important prerequisites for developing and analysi ng methods for the fast solution of the resulting linear systems (Sect . 4). These methods are based on compressing the stiffness matrices re lative to wavelet bases for the given multiresolution analysis. Such a compression technique has been proposed in [2] where, however, only c lassical 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(log N)(b)), where N is the number of unknowns and b greater than or equal to 0 is some real number. In Sect. 5 the theoretical results are conf irmed by new numerical experiments for the exterior Dirichlet problem for the Helmholtz equation.