A localized orthogonal decomposition method for semi-linear elliptic problems

Abstract

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H log(H −1 ) where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H 1 -error with respect to the coarse mesh size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.

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.…”

“…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

“…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.…”

Domain Decomposition Methods in Science and Engineering XXIII

“…We refer to Henning et al (2014) and for a detailed discussion. We refer to Henning et al (2014) and for a detailed discussion.…”

“…Using classical homogenization as a guideline, these modifications are obtained from local auxiliary problems (Abdulle et al, 2012;Efendiev and Hou, 2009;Hughes et al, 1998). These restrictions were overcome in a recent paper by that provides quasioptimal energy and L 2 error estimates without any additional assumptions on periodicity and scale separation (Henning et al, 2014;. These restrictions were overcome in a recent paper by that provides quasioptimal energy and L 2 error estimates without any additional assumptions on periodicity and scale separation (Henning et al, 2014;.…”