Oversampling for the Multiscale Finite Element Method

Abstract

This paper reviews standard oversampling strategies as performed in the Multiscale Finite Element Method (MsFEM). Common to those approaches is that the oversampling is performed in the full space restricted to a patch but including coarse finite element functions. We suggest, by contrast, to perform local computations with the additional constraint that trial and test functions are linear independent from coarse finite element functions. This approach re-interprets the Variational Multiscale Method in the context of computational homogenization. This connection gives rise to a general fully discrete error analysis for the proposed multiscale method with constrained oversampling without any resonance effects. In particular, we are able to give the first rigorous proof of convergence for a MsFEM with oversampling.

Eliminating the pollution effect in Helmholtz problems by local subscale correction

“…This application and the generalization of the method are very natural and straight forward. Though this case is not yet covered, previous work [MP14b,EGMP13,HMP14b,HP13,HMP14a] plus the analysis of this paper strongly indicate the potential of the method to treat high oscillations or jumps in the PDE coefficients and the pollution effect in one stroke. The remaining part of the paper is outlined as follows.…”

“…The only technical issue is that the space V h is not closed under multiplication by cut-off functions used in the proofs of Theorem 4.6, Lemma 5.1, and Theorem 5.2. This requires minor modifications as they have already been applied successfully in previous papers [MP14b,HP13,HMP14a]. To begin with, let all cut-off functions η be replaced by their nodal interpolation I nodal H η on the coarse mesh T H .…”

“…Numerical homogenization (or upscaling) refers to a class of multiscale methods for the efficient approximation on coarse meshes that do not resolve the coefficient oscillations. A novel method for this problem was recently introduced in [MP14b] and further generalized in [EGMP13,HMP14b,HP13,HMP14a]. The method is based on localizable orthogonal decompositions (LOD) into a low-dimensional coarse space (where we are looking for the approximation) and a high-dimensional remainder space.…”

Computation of eigenvalues by numerical upscaling

“…We refer to [MP14] for details and proofs. The modified localization procedures from [HP13] and [HMP14a] with improved accuracy and stability properties might as well be applied.…”

“…In all experiments, we focus on the case without localization. The localization (as discussed in Section 6.1) has been studied extensively for the linear problem in [MP14,HP13,HM14,HMP14a] and for semi-linear problems in [HMP14c]. In the present context of eigenvalue approximation, we are interested in observing the enormous convergence rate which is 4 or higher for the eigenvalues.…”