Equilibrated residual error estimates are p-robust

Braess, Dietrich; Pillwein, Veronika; Schöberl, Joachim

doi:10.1016/j.cma.2008.12.010

Cited by 141 publications

(227 citation statements)

References 17 publications

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“…We now estimate the dual residual norm Res(u ih , λ ih ) −1 , for u ih ∈ V a piecewise polynomial of degree p ≥ 1 and λ ih ∈ R. For the upper bound, following [49,19,8,22] and [21,47,46] for inexact solvers, see also the references therein, we introduce an equilibrated flux reconstruction. This is a vector field σ ih constructed from the local residual of (u ih , λ ih ) by solving patchwise mixed finite element problems such that will allow us to construct r ih ∈ X h ⊂ V leading to a lower bound on Res(u ih , λ ih ) −1 .…”

Section: Dual Norm Of the Residual Equivalencesmentioning

confidence: 99%

“…This question is usually tackled via a posteriori error estimates. For elliptic source problems such as the Laplace one, conclusive answers are today given by, in particular, the theory of equilibrated fluxes following Prager and Synge [49], see Destuynder and Métivet [19], Braess et al [8], Ern and Vohralík [22], and the references therein. The structure of the Laplace eigenvalue problem appears rather richer in comparison with the elliptic source case.…”

Section: Introductionmentioning

confidence: 99%

“…From the sign assumption (u i , u ih ) ≥ 0, employing u i = u ih = 1, 8) so that C ih e ih ≤ 2 r (ih) 2 , i.e., (3.2). Note that inspecting more closely the quadratic inequality (3.7), the improved bound e ih ≤ 2 − 4 − 2α 2 ih ( √ 2-times better for e ih approaching zero) follows under condition r (ih) 2 < C ih that we prefer to avoid.…”

mentioning

confidence: 99%

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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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Section: Dual Norm Of the Residual Equivalencesmentioning

confidence: 99%

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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show abstract

“…The two-energies principle and the theorem of Prager and Synge have been used for efficient a posteriori error estimates of other elliptic problems; see, e.g., [6,8]. Since the principle was used successfully in the preceding sections for a priori error estimates, it is expected to be also a good candidate for deriving a posteriori error estimates of the model error.…”

Section: A Posteriori Estimates Of the Model Errormentioning

confidence: 99%

On the Justification of Plate Models

Braess

Sauter

Schwab

2010

J Elast

Self Cite

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In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff-Love and ReissnerMindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern [Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Rational Mech. Anal. 4, 145-152 (1959)] and is based on the two-energies principle of Prager and Synge. This was half a centenium ago.We will derive the Kirchhoff-Love model based on Morgenstern's ideas in a rigorous way (including the proper treatment of boundary conditions). It provides insights a) for the relation of the (1, 1, 0)-model with the (1, 1, 2)-model that differ by a quadratic term in the ansatz for the third component of the displacement field and b) for the rôle of the shear correction factor. A further advantage of the approach by the two-energy principle is that the extension to the ReissnerMindlin plate model becomes very transparent and easy. Our study includes plates with reentrant corners.

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“…We remark, however, that the p-suboptimality is less pronounced in energy norm lower bounds. We also mention the alternative approach presented in Braess et al (2009) in the context of spectral methods.…”

Section: P Houston and T P Wihlermentioning

confidence: 99%

Second-order elliptic PDEs with discontinuous boundary data

Houston

Wihler

2011

IMA Journal of Numerical Analysis

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We consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well defined in the standard H 1 −H 1 setting we introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive (enhanced) L 2 -norm upper and local lower a posteriori error bounds. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h-and the hp-version setting. Indeed, in the latter case, exponential convergence of the error is attained as the mesh is adaptively refined.

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Equilibrated residual error estimates are p-robust

Cited by 141 publications

References 17 publications

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

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

On the Justification of Plate Models

Second-order elliptic PDEs with discontinuous boundary data

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