Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method

Carstensen, Carsten; Ern, Alexandre; Puttkammer, Sophie

doi:10.1007/s00211-021-01228-1

Cited by 10 publications

(5 citation statements)

References 34 publications

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“…Numerical experiments in [CP21,CEP21] show an asymptotic convergences of AFEM4EVP with θ " 0.5 even for coarse initial triangulation and confirm the optimal convergence rates of Theorem 1.1 even for one example with a multiple eigenvalue. The extension to eigenvalue clusters requires an algorithm from [Gal15b,DHZ15,BGGG17].…”

Section: Numerical Experimentssupporting

confidence: 62%

“…The maximal mesh-size h max enters as an explicit parameter and this can be non-effective for an imperative adaptive mesh-refinement. This has recently motivated the design of extra-stabilized nonconforming finite element eigensolvers for m " 1, 2 that directly compute GLB under moderate mesh-size restrictions and allow an efficacious adaptive mesh-refinement [CZZ20, CEP21,CP21]. The striking superiority of those adaptive schemes has been displayed in numerical experiments in [CEP21,CP21] and motivates the mathematical analysis of optimal convergence rates in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…This has recently motivated the design of extra-stabilized nonconforming finite element eigensolvers for m " 1, 2 that directly compute GLB under moderate mesh-size restrictions and allow an efficacious adaptive mesh-refinement [CZZ20, CEP21,CP21]. The striking superiority of those adaptive schemes has been displayed in numerical experiments in [CEP21,CP21] and motivates the mathematical analysis of optimal convergence rates in this paper. This appears to be the first method that combines the localization of eigenvalues as GLB with their efficient approximation.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Carstensen¹,

Puttkammer²

2022

Preprint

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Guaranteedlower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix-Raviart (m " 1) or Morley (m " 2) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated L 2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

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Section: Numerical Experimentssupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Carstensen¹,

Puttkammer²

2022

Preprint

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“…The higher-order Crouzeix-Raviart finite element schemes are complicated at least in 3D [23] and then the HHO methodology is an attractive alternative even for simplicial triangulations with partly unexpected advantages like the computation of higher-order guaranteed eigenvalue bounds [15]. Higher convergence rates rely on an appropriate adaptive mesh-refining algorithm and hence stabilization-free a posteriori error estimators are of particular interest.…”

Section: Further Contributions and Outlinementioning

confidence: 99%

Stabilization-free HHO a posteriori error control

Bertrand¹,

Carstensen²,

Gräßle³

et al. 2022

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The known a posteriori error analysis of hybrid high-order methods (HHO) treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks. Keywords hybrid high-order • a posteriori • guaranteed upper error bounds • adaptive mesh refinement • equilibration • stabilization-free • computational comparisons This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) in the Priority Program 1748 Reliable simulation techniques in solid mechanics. Development of nonstandard discretization methods, mechanical and mathematical analysis under the projects BE 6511/1-1 and CA 151/22-2 as well as the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 891734). The third author is also supported by the Berlin Mathematical School.

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“…We refer the reader to the seminal contributions in Vainikko [32,33], Bramble and Osborn [4], Strang and Fix [31], Osborn [28], Descloux et al [19,20], Babuška and Osborn [2], and to the more recent reviews in [3,21]. The approximation of elliptic spectral problems has also been studied by means of mixed finite element methods [12,27,26], discontinuous Galerkin methods [1,23], hybridizable discontinuous Galerkin methods [14,24], hybrid high-order methods [10,13], and virtual element methods [22]. All of these methods deliver optimally-convergent approximations, but since the eigenfunctions become more and more oscillatory in the upper part of the spectrum, the approximation is accurate only in the lower part of the spectrum.…”

Section: Introductionmentioning

confidence: 99%

SoftFEM: revisiting the spectral finite element approximation of second-order elliptic operators

Deng,

Ern

2020

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We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts, but it brings here a negative contribution. Since it reduces the stiffness of the problem, the resulting approximation technique is called softFEM. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. We derive a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form. Then we prove that softFEM delivers the same optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We next compare the discrete eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a sensible choice for the softness parameter. With this choice, the stiffness reduction ratio scales linearly with the polynomial degree. Various numerical experiments illustrate the benefits of using softFEM over Galerkin FEM.

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Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method

Cited by 10 publications

References 34 publications

Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Stabilization-free HHO a posteriori error control

SoftFEM: revisiting the spectral finite element approximation of second-order elliptic operators

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