A multilevel preconditioning for generalized finite element method problems on unstructured simplicial meshes

Abstract

This paper is on the efficient solution of linear systems arising in discretizations of second order elliptic PDEs by a generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We propose an efficient preconditioner by using auxiliary (fictitious) space techniques and an additive preconditioner for the auxiliary space problems. We also prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh parameters.