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

Abstract

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 precomputed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional inter-element communication.

On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems Without Oversampling Techniques

“…In practice, avoiding oversampling strategy allows saving computational cost, and this also corroborates well empirical observations [14]. Due to the local estimates in Proposition 4.1 and Lemma 4.3, we are able to derive a global estimate in Proposition 4.2 that is the much needed results for analyzing many multiscale methods [19,6,25,22], cf. Remark 4.2.…”

“…Due to the disparity of scales, the classical numerical treatment becomes prohibitively expensive and even intractable for many multiscale applications. Nonetheless, motivated by the broad spectrum of practical applications, a large number of multiscale model reduction techniques, e.g., multiscale finite element methods (MsFEMs), heterogeneous multiscale methods (HMMs), variational multiscale methods, flux norm approach, generalized multiscale finite element methods (GMsFEMs) and localized orthogonal decomposition (LOD), have been proposed in the literature [18,11,19,6,12,25,22] over the last few decades. They have achieved great success in the efficient and accurate simulation of heterogeneous problems.…”

Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

“…They therefore contain no information on initial states and one cannot take advantage of knowledge on how "information" is transported by the dynamics inherent to the model. Multiscale methods for stationary problems [3,20,21,26,28] therefore are conceptually different. To the best of our knowledge existing methods for transient problems usually aim at different regimes, e.g., see [16,27] and make some (legitimate) assumptions of the flow field specific to the application that motivated the authors.…”

Multiscale stabilization for convection–diffusion equations with heterogeneous velocity and diffusion coefficients

“…The least squares approaches [7,49,8,15] can be used to achieve stability in the natural norm. We also note that a stabilization technique based on the variational multiscale method is presented in [51]. Our goal is to design procedures for constructing test functions that are localizable and give good stability, that work well for large Peclet numbers.…”