A reduced basis localized orthogonal decomposition

Abdulle, Assyr; Henning, Patrick

doi:10.1016/j.jcp.2015.04.016

Cited by 31 publications

(42 citation statements)

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“…The P1-FE approximation (•) on a uniform mesh of width h = 2 −7 > 6 · (wave length) fails to approximate u κ due to the accumulation of phase errors. lems [MP15b,HMP14a,MP15c], parabolic problems [MP15a], wave propagation [AH14, Pet14, GP15] and parametric problems [AH15].…”

Section: Introductionmentioning

confidence: 99%

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

Peterseim

2016

Lecture Notes in Computational Science and Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

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

Peterseim

2016

Lecture Notes in Computational Science and Engineering

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“…In order to address parameterized multiscale problems the local approximation spaces are for instance spanned by eigenfunctions of an eigenvalue problem on the space of harmonic functions in [28], generated by solving the global parameterized PDE and restricting the solution to the respective subdomain in [70,3], or enriched in the online stage by local solutions of the PDE, prescribing the insufficient RB solution as Dirichlet boundary conditions in [70,3]. Apart from that the RB method has also been used in the context of multiscale methods for example in [69,40,1].…”

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confidence: 99%

Randomized Local Model Order Reduction

Buhr¹,

Smetana²

2018

SIAM J. Sci. Comput.

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In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babuška and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. Moreover, the adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.Key words. localized model order reduction, randomized linear algebra, domain decomposition methods, multiscale methods, a priori error bound, a posteriori error estimation AMS subject classifications. 65N15, 65N12, 65N55, 65N30, 65C20, 65N25

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“…• Mh: Fine-mesh Mass-matrix, see definition 3.4. In many cases it can be convinient to also compute the mass matrix element wise, see 3.3, and then compute the global matrix as given in (10). • Ph: Projection matrix P h from the coarse-mesh Lagrange space to the fine-mesh Lagrange space, see equation (14).…”

Section: Numerical Examplesmentioning

confidence: 99%

Efficient implementation of the localized orthogonal decomposition method

Engwer

Henning

Målqvist

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems. the field of multiscale partial differential equations including rough polyharmonic splines [41], iterative numerical homogenization [42], and gamblets [43]. While previous works focused on the numerical analysis of the method, this paper aims at the detailed explanation of how the method can be algorithmically realized. We give detailed explanations on how the method works on an algebraic level. The results may as well be useful for implementing related multiscale methods. PreliminariesIn this section we recall the Localized Orthogonal Decomposition (LOD) for finite element spaces. The decomposition is always with respect to a linear elliptic part of the differential operator. Computational domain and boundaryFor the rest of the paper, we consider a bounded polygonal domain Ω ⊂ R d . The boundary ∂Ω is divided into two parts Γ D and Γ N . On Γ D we prescribe a Dirichlet boundary condition and on Γ N we prescribe a Neumann boundary condition. We have Γ D ∪ Γ N = ∂Ω and we assume Γ D = ∅. With that, we define the spacewhere v |Γ D = 0 is understood in the sense of traces. Elliptic differential operatorSubsequently we consider the following differential operator. Let κ ∈ L ∞ (Ω, R d×d ) denote a matrix-valued, symmetric, possibly highly varying and heterogeneous coefficient with uniform spectral bounds γ min > 0 and γ max ≥ γ min , σ(κ(x)) ⊂ [γ min , γ max ] for almost all x ∈ Ω.This coefficient defines a scalar product A(·, ·) on H 1 Γ D (Ω) that is given by Meshes and spacesWe wish to discretize a problem that is associated with A(·, ·). Then the discretization is constrained by the diffusion coefficient κ, in the sense that variations of κ must be resolved by the computational mesh. We call such a discretization a fine scale discretization. In addition to this, we have a second discretization on a coarse scale. The coarse mesh is arbitrary and no more related to A(·, ·). It contains elements of maximum diameter H > 0. The fine mesh consists of elements of maximum diameter h < H. Let T H , T h denote the corresponding simplicial or quadrilateral subdivisions of Ω into (closed) conforming shape regular simplicial elements or conforming shape regular quadrilateral elements, i.e.,Ω =We assume that T h is a regular, possibly non-uniform, mesh refinement of T H . Furthermore we also assume that T H and T h are shape-regular in the sense that there exists a positive constant c 0 such that max maxand regular in the sense that any two elements are either disjoint, share exactly one face, share exactly one edge, or share exactly one ve...

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A reduced basis localized orthogonal decomposition

Cited by 31 publications

References 50 publications

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

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

Randomized Local Model Order Reduction

Efficient implementation of the localized orthogonal decomposition method

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