Finite element approximation of eigenvalue problems

Boffi, Daniele

doi:10.1017/s0962492910000012

Cited by 423 publications

(402 citation statements)

References 136 publications

(195 reference statements)

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“…In the subsequent paragraphs we will present error bounds for the approximate eigenvalues and eigenfunctions based on the variational techniques from [SF73] (which are based on [BdBSW66] on their part); see also [Bof10].…”

Section: Error Analysismentioning

confidence: 99%

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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Section: Error Analysismentioning

confidence: 99%

Computation of eigenvalues by numerical upscaling

Målqvist

Peterseim

2014

Numer. Math.

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“…In extension of some classical results, see [26,2,7,58], we establish in this section generic equivalence results between the following three quantities: the i-th eigenvalue error ∇u ih 2 − λ i , which can potentially be negative, the square of the i-th eigenvector energy error ∇(u i − u ih ) 2 , and the square of the dual norm of the residual Res(u ih , λ ih ) 2 −1 . These equivalences may for the moment contain uncomputable terms like the eigenvalues λ i−1 , λ i , λ i+1 or the Riesz representation norm r (ih) , but all such terms will be removed later.…”

Section: Generic Equivalencesmentioning

confidence: 73%

“…Gilbarg and Trudinger [26], Babuška and Osborn [2], Boffi [7], or Strang and Fix [58], u k , k ≥ 1, form a countable orthonormal basis of L 2 (Ω) consisting of vectors from V , whereas 0 < λ 1 < λ 2 ≤ λ 3 ≤ . .…”

Section: The Laplace Eigenvalue Problemmentioning

confidence: 99%

“…For conforming finite elements, relying on the a priori error estimates resumed in Babuška and Osborn [2] and Boffi [7], see also the references therein, a posteriori error estimates have been obtained by Verfürth [60], Maday and Patera [43], Larson [38], Heuveline and Rannacher [31], Durán et al [20], Grubišić and Ovall [29], Rannacher et al [50], andŠolín and Giani [56], see also the references therein. These estimates, though, systematically contain uncomputable terms, typically higher order on fine enough meshes.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

Cancès¹,

Dusson²,

Maday³

et al. 2017

SIAM J. Numer. Anal.

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This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue. They are valid without restrictions on the computational mesh or on the approximate eigenvector; we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones, which can be checked in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hypotheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory, and no convexity/regularity assumption on the computational domain/exact eigenvector(s) is needed. Two improvements of the multiplicative constant appearing in our estimates are presented. First, it is reduced by a fixed factor under an explicit, a posteriori calculable condition on the mesh and on the approximate eigenvector-eigenvalue pair. Second, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, the multiplicative factor can be brought to the optimal value of one. Inexact algebraic solvers are taken into account; the estimates are valid on each iteration and can serve for the design of adaptive stopping criteria. The application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided, along with a numerical illustration on a set of test problems.

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“…In 1980s, Nédélec [10,11] proposed the edge elements, which can preserve this actual physical property of electric field E. Therefore edge elements are very well suited for approximating electric field E. The review of edge element method has been given in [12]. For the Maxwell's eigenvalue problem in isotropic media, with the edge element method there are no spurious modes with nonzero eigenvalues, but the number of spurious modes with nonphysical zero eigenvalue is equal to the number of interior nodes inside the computational domain [13,14]. Therefore, in order to remove these second-kind spurious modes with nonphysical zero eigenvalues, we must make the eigenfunction have the property of divergence-free condition.…”

Section: Introductionmentioning

confidence: 99%

Mixed Finite Element Method for 2d Vector Maxwell's Eigenvalue Problem in Anisotropic Media

Jiang¹,

Liu²,

Tang³

et al. 2014

PIER

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Abstract-It is well known that the conventional edge element method in solving vector Maxwell's eigenvalue problem will lead to the presence of nonphysical zero eigenvalues. This paper uses the mixed finite element method to suppress the presence of these nonphysical zero eigenvalues for 2D vector Maxwell's eigenvalue problem in anisotropic media. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on employing the first-order edge element basis functions to expand the electric field and linear nodal element basis functions to expand the Lagrangian multiplier. Our numerical experiments show that this method can successfully remove all nonphysical zero and nonzero eigenvalues. We verify that when the cavity has a connected perfect electric boundary, then there is no physical zero eigenvalue. Otherwise, the number of physical zero eigenvalues is one less than the number of disconnected perfect electric boundaries.

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Finite element approximation of eigenvalue problems

Cited by 423 publications

References 136 publications

Computation of eigenvalues by numerical upscaling

Computation of eigenvalues by numerical upscaling

Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

Mixed Finite Element Method for 2d Vector Maxwell's Eigenvalue Problem in Anisotropic Media

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