Variational Multiscale Stabilization and the Exponential Decay of Fine-Scale Correctors

Abstract

This paper reviews the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L 2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.

Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering

“…In the numerical experiments of [11], a variant of that method appeared attractive where only the test functions are modified while standard finite element functions are used as trial functions. In this paper, we analyze that method and reformulate it as a stabilized Q 1 method in the spirit of the variational multiscale method [13][14][15][16][17]. The method employs standard Q 1 finite element trial functions on a grid G H with mesh-size H .…”

Computational multiscale methods for linear poroelasticity with high contrast

“…Another technique to solve multiscale problems is the Localized Orthogonal Decomposition method (LOD), cf. [13,16,17,23,25]. This method was applied in [22] to linear heterogeneous thermoelasticity and optimal first-order convergence of the fully discretized system based on LOD and an implicit Euler discretization in time was proven.…”

“…Therein, it is shown that this method has a convergence rate proportional to the coarse grid size, which remains valid even in the presence of high contrast provided that sufficiently many basis functions are selected. The approach makes use of the ideas of localization [20,21,23,25] and oversampling to compute multiscale basis functions in some oversampled subregions with the aim to obtain an appropriate orthogonality condition.…”

Online basis construction for goal-oriented adaptivity in the generalized multiscale finite element method

“…The direct simulation of multiscale PDEs with accurate resolution can be costly as a relatively fine mesh is required to resolve the coefficients, leading to a prohibitively large number of degrees of freedom (DOF), a high percentage of which may be extraneous. Recently, these computational challenges have been addressed by the development of efficient model reduction techniques such as numerical homogenization methods [17,18,27,31,32] and multiscale methods [5,6,7,22,33,34]. These methods have been shown to reduce the computational cost of the simulation, for instance approximating u of (1).…”