Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

Ainsworth, Mark; Rankin, Richard

doi:10.1016/j.cma.2011.01.017

Cited by 13 publications

(14 citation statements)

References 30 publications

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“…where C K and C ,K are given by (14). Then, applying the Cauchy-Schwarz's inequality to (24), (25) and recalling the definition of the error indicator Á K in (10), we arrive at…”

Section: Proof Of Theorem 51mentioning

confidence: 99%

Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions

Ainsworth

Allendes

Barrenechea

et al. 2013

Numerical Methods in Fluids

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SUMMARYFully computable upper bounds are developed for the discretisation error measured in the natural (energy) norm for convection–reaction–diffusion problems in three dimensions. The upper bounds are genuine upper bounds in the sense that the numerical value of the estimated error exceeds the actual numerical value of the true error regardless of the coarseness of the mesh or the nature of the data for the problem. All constants appearing in the bounds are fully specified. Examples show the estimator to be reliable and accurate even in the case of complicated three‐dimensional problems. Copyright © 2013 John Wiley & Sons, Ltd.

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“…where C K and C ,K are given by (14). Then, applying the Cauchy-Schwarz's inequality to (24), (25) and recalling the definition of the error indicator Á K in (10), we arrive at…”

Section: Proof Of Theorem 51mentioning

confidence: 99%

Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions

Ainsworth

Allendes

Barrenechea

et al. 2013

Numerical Methods in Fluids

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“…It turns out to be quite difficult to design finite elements in H(div, Ω ; S) which preserve both the symmetry and the H(div)-conformity. The first successful discretization use the so-called composite elements, see the low-order composite elements in Johnson & Mercier (1978) and the high-order composite elements in Arnold et al (1984), both defined on triangular and quadrilateral meshes; see also a low-order composite element on tetrahedral meshes in Ainsworth & Rankin (2011). While these methods can be efficiently implemented via hybridization, leading to an symmetricpositive-definite linear system, their main drawback is that their basis functions are usually quite hard to construct, especially for the high-order methods in Arnold et al (1984).…”

Section: Introductionmentioning

confidence: 99%

Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions

Cockburn

2017

IMA Journal of Numerical Analysis

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We propose a new tool, which we call M M M-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an M M M-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit M M M-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.

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“…The second one follows the same approach, but imposing the symmetry only weakly and using the Arnold-Falk-Winther (AFW) mixed finite element spaces [2]. Element-wise reconstructions of equilibrated stress tensors from local Neumann problems can be found in [13,1,12], whereas direct prescription of the degrees of freedom in the AW finite element space is considered in [14].…”

Section: Introductionmentioning

confidence: 99%

Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We present an a posteriori error estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions: one using Arnold-Winther mixed finite element spaces providing a symmetric stress tensor, and one using Arnold-Falk-Winther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

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Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

Cited by 13 publications

References 30 publications

Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions

Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions

Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions

Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects

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