On solving operator equations by Galerkin's method with Gabor frame

Abstract

‎This paper deals with solving boundary value problems by Galerkin's method in which we use Gabor frames as trial and test functions‎. ‎We show that‎, ‎the preconditioned stiffness matrix resulted by discretization is compressible and its sparsity‎ ‎pattern involves a bounded polyhedron structure‎. ‎Moreover‎, ‎we introduce an adaptive Richardson iterative method to‎ ‎solve the resulting system and we also show that by choosing a suitable smoothing parameter‎, ‎the method would be convergent‎.