Localization of elliptic multiscale problems

Målqvist, Axel; Peterseim, Daniel

doi:10.1090/s0025-5718-2014-02868-8

Cited by 410 publications

(604 citation statements)

References 22 publications

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“…The result in the decay of P f φ z in [MP14] can be expressed as follows. For all vertices z ∈ N H and for all k ∈ N, it holds…”

Section: Practical Aspectsmentioning

confidence: 99%

“…Due to the exponential decay, the very weak condition k ≈ | log H| implies that the perturbation of the ideal method due to this truncation is of higher order and the estimates in Theorems 5.2 and 5.3 remain valid. 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.…”

Section: Practical Aspectsmentioning

confidence: 99%

“…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.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Computation of eigenvalues by numerical upscaling

Målqvist

Peterseim

2014

Numer. Math.

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Abstract. We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on twoscale decompositions of H 1 0 (Ω) by means of a certain Clément-type quasi-interpolation operator.

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“…The result in the decay of P f φ z in [MP14] can be expressed as follows. For all vertices z ∈ N H and for all k ∈ N, it holds…”

Section: Practical Aspectsmentioning

confidence: 99%

Section: Practical Aspectsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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Computation of eigenvalues by numerical upscaling

Målqvist

Peterseim

2014

Numer. Math.

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“…This paper adapts the multiscale method of [MP14b] to cure pollution in the numerical approximation of the Helmholtz problem. To deal with the lack of hermitivity we will propose a Petrov-Galerkin version of the method (although this is not essential).…”

Section: Introductionmentioning

confidence: 99%

Eliminating the pollution effect in Helmholtz problems by local subscale correction

Peterseim

2016

Math. Comp.

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115

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Abstract. We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number κ in bounded domains in R d . The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale correction in the spirit of numerical homogenization. The precomputation of the correction involves the solution of coercive cell problems on localized subdomains of size ℓH; H being the mesh size and ℓ being the oversampling parameter. If the mesh size and the oversampling parameter are such that Hκ and log(κ)/ℓ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to H; pollution effects are eliminated in this regime.

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“…The underlying technique is well-established in the context of numerical homogenisation [1]. The reduced space V H allows for an improved inverse inequality that decouples the time step from the minimal mesh size and turns the leapfrog into a feasible numerical scheme also on adaptive spatial meshes.…”

Section: The Wave Equation On Domains With Re-entrant Cornersmentioning

confidence: 99%

Relaxing the CFL condition for the wave equation on adaptive meshes

Peterseim¹,

Schedensack²

2016

Proc Appl Math and Mech

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The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. A simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities. We consider the discretisation of the wave equation, with (possibly) re-entrant corners. This typically reduces the regularity of the solution and leads to u(t) ∈ H 2 (Ω). To reveal optimal convergence rates, non-uniform mesh refinement in space proves advantageous for the wave equation [2]. Given a discrete space V H , the leapfrog scheme seeks a solution (u n H ) n=0,...,N with u n H ∈ V H such that for all n = 2, . . . , N and allAs usual for explicit time discretisation schemes, the numerical stability is conditional and guaranteed only under the sharp Courant-Friedrichs-Lewy (CFL) condition. In the present context of linear finite elements, it bounds the time step size by the minimal mesh size of the spatial mesh ∆t h min .While on (quasi-)uniform meshes the admissible choice ∆t ≈ h min ≈ h max is considered as a natural balance of space and time discretisation, the CFL condition is not at all compatible with non quasi-uniform meshes in the sense that the efficiency of adaptive mesh refinement in space causes tiny time steps that destroy the overall complexity. Essentially, the CFL condition forbids any type of spatial adaptivity.It is proved in [3] that the restriction of the time step by the minimal spatial mesh size can easily be removed by projecting the adaptive finite element space to some subspace V H with similar (optimal) approximation properties for weak solutions of the wave equation. The underlying technique is well-established in the context of numerical homogenisation [1]. The reduced space V H allows for an improved inverse inequality that decouples the time step from the minimal mesh size and turns the leapfrog into a feasible numerical scheme also on adaptive spatial meshes. Spatial reductionWe consider a quasi-uniform shape regular triangulation T H with (maximal) mesh size H and some (possibly) non-quasiuniform shape regular triangulation and refinement T h of T H with corresponding standard P 1 -FEM spaces S 1 0 (T H ) and S 1 0 (T h ). We assume that T h leads to an optimal approximation property in the sense thatThe construction of the generalised finite element space is based on a projective quasi-interpolation operator I H : V → S 1 0 (T H ) with approximation and stability properties

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Localization of elliptic multiscale problems

Cited by 410 publications

References 22 publications

Computation of eigenvalues by numerical upscaling

Computation of eigenvalues by numerical upscaling

Eliminating the pollution effect in Helmholtz problems by local subscale correction

Relaxing the CFL condition for the wave equation on adaptive meshes

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