Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions

Abstract

© 2017 The authors. We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed finescale correctors. The exponential decay of these correctors and their localization to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, are rigorously justified. Under moderate assumptions, this stabilization guarantees stability and a quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional interelement communication.