An analysis of a class of variational multiscale methods based on subspace decomposition

Kornhuber, Ralf; Peterseim, Daniel; Yserentant, Harry

doi:10.1090/mcom/3302

Cited by 108 publications

(100 citation statements)

References 16 publications

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“…Quantitative localization results for this particular case have been derived in [1] using classical results from domain decomposition and the convergence theory of iterative solvers, cf. [4,5]. From this theory we know that localization appears if the potential exhibits disorder and satisfies β ε −2 .…”

Section: Numerical Resultsmentioning

confidence: 99%

Localization studies for ground states of the Gross‐Pitaevskii equation

Altmann

Peterseim

Varga

2018

Proc Appl Math and Mech

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For oscillatory high-amplitude potentials it is known that the linear Schrödinger operator leads to exponentially localized ground states. This localization result can be quantified explicitly in terms of geometric parameters and the degree of disorder within the potential. In the present paper we study the influence of the amplitude and the importance of the periodicity in the nonlinear setting of the Gross-Pitaevskii equation which models ultracold bosonic gases. Gross-Pitaevskii equationQuantum-physical processes related to ultracold bosonic gases -so-called Bose-Einstein condensates -can in certain regimes be mathematically described by the Gross-Pitaevskii equation [7]. The corresponding eigenvalue problem is then related to the formation of ground states and excited states of such condensates. For oscillatory and disordered potentials surprising phenomena such as Anderson localization occur. We are particularly interested in such localization properties of the ground state, which is the minimal-energy solution of the problemFor simplicity, we consider homogeneous Dirichlet boundary conditions, i.e., u ∈ H 1 0 (Ω). The parameter δ ≥ 0 controls the non-linearity whereas V is a high-amplitude potential that oscillates on a small scale ε. Moreover, the potential V ∈ L ∞ (D) remains within the boundsProperties of the ground state are discussed in [6]. The aim of this paper is to study the influence of the non-linearity (in form of the parameter δ) and the amplitude of the potential (parameter β) on the possible localization of the ground state. Numerical resultsFor the numerical experiments, we introduce three prototypical potentials. The first potential V 1 is periodic and oscillates continuously between the values α and β. Since the influence of the lower bound is negligible, we fix α = 1 throughout all experiments. In two space dimensions the first potential reads V 1 (x, y) := α + β−α 2 1 + sin(2πx/ε) sin(2πy/ε) .The second and third potential are both piecewise constant (on an ε-mesh). The potential V 2 takes independently the value α or β with probability 1 2 each (coin toss). V 3 takes independent random values uniformly distributed in the interval [α, β], cf. Fig. 1. (a) periodic potential V1 (b) checkerboard potential V2 (c) random potential V3 Fig. 1: Realizations of the three potentials V1, V2, and V3.

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Section: Numerical Resultsmentioning

confidence: 99%

Localization studies for ground states of the Gross‐Pitaevskii equation

Altmann

Peterseim

Varga

2018

Proc Appl Math and Mech

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“…see also [21,20] for an alternative constructive proof. Similar to the estimates in Section 3.1, we obtain…”

Section: Localization Of Correctorsmentioning

confidence: 99%

Explicit computational wave propagation in micro-heterogeneous media

Maier

Peterseim

2018

Bit Numer Math

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Explicit time stepping schemes are popular for linear acoustic and elastic wave propagation due to their simple nature which does not require sophisticated solvers for the inversion of the stiffness matrices. However, explicit schemes are only stable if the time step size is bounded by the mesh size in space subject to the so-called CFL condition. In micro-heterogeneous media, this condition is typically prohibitively restrictive because spatial oscillations of the medium need to be resolved by the discretization in space. This paper presents a way to reduce the spatial complexity in such a setting and, hence, to enable a relaxation of the CFL condition. This is done using the Localized Orthogonal Decomposition method as a tool for numerical homogenization. A complete convergence analysis is presented with appropriate, weak regularity assumptions on the initial data.

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“…In numerical experiments with a Cantor-type geometry, we found the theoretically predicted behavior of (finite element approximationsũ K of) u K . We also observed that the convergence rates of our iterative scheme appear to be robust with respect to increasing K. Theoretical justification and extensions to model reduction in the spirit of [25,28] are subject of current research.…”

Section: Introductionmentioning

confidence: 71%

“…For a numerical illustration, we consider this example in d = 2 space dimensions with c = 1, f ≡ 1, A ≡ 1, and the geometrical parameter C K = 2 K−1 . Note that the ⋅ norm in H (cf (25)(27)…”

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confidence: 99%

Fractal Homogenization of Multiscale Interface Problems

Heida¹,

Kornhuber²,

Podlesny³

2020

Multiscale Model. Simul.

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Inspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of non-separating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into L 2 and H s , s < 1 2. We show existence and uniqueness of the solution of the asymptotic limit problem and exponential convergence of the approximating finite-scale solutions. Computational experiments involving a related numerical homogenization technique illustrate our theoretical findings.

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An analysis of a class of variational multiscale methods based on subspace decomposition

Cited by 108 publications

References 16 publications

Localization studies for ground states of the Gross‐Pitaevskii equation

Localization studies for ground states of the Gross‐Pitaevskii equation

Explicit computational wave propagation in micro-heterogeneous media

Fractal Homogenization of Multiscale Interface Problems

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