Fully Computable Bounds for the Error in Nonconforming Finite Element Approximations of Arbitrary Order on Triangular Elements

Ainsworth, Mark; Rankin, Richard

doi:10.1137/07070838x

Cited by 30 publications

(28 citation statements)

References 15 publications

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“…The nonconforming finite element method for (2.2), cf. Stoyan and Baran [71] or [9], reads: find u h ∈ V h such that…”

Section: Nonconforming Finite Elementsmentioning

confidence: 99%

“…Additionally, relying on the fact that the lifting l maps onto piecewise constant functions, 9) where T e stands for the (one or two) elements sharing the edge e and C κT uniformly bounds he |K| 1/2 and only depends on the mesh-regularity parameter κ T . Finally, 10) where C only depends on the mesh-regularity parameter κ T and E + K := {e ∈ E h |∃a ∈ V K , ∃K ′ ∈ T a , e ∈ E K ′ }, so that (4.5) combined with (4.10) yields polynomial-degree-robust local efficiency for the same error measure as in (4.7).…”

Section: Potential Reconstruction For the Nonsymmetric And Incompletementioning

confidence: 99%

“…Different flux equilibrations exist and tight links hold between them, see [46]. Higher-order methods have been treated by Ainsworth and Rankin [9], and a survey and a computational comparison in the lowest-order case can be found in Carstensen and Merdon [29]. For the discontinuous Galerkin method, first guaranteed upper bounds by locally equilibrated fluxes and H 1 (Ω)-conforming reconstructed potentials have been obtained by Ainsworth [3], Kim [52,53], Cochez-Dhondt and Nicaise [32], Ainsworth and Rankin [10], and [41,43,42], see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Ern¹,

Vohralı́k²

2015

SIAM J. Numer. Anal.

163

196

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We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.Key words: a posteriori error estimate, equilibrated flux, unified framework, robustness, polynomial degree, conforming finite element method, nonconforming finite element method, discontinuous Galerkin method, mixed finite element method IntroductionA posteriori error estimates in the conforming finite element setting have already received a large attention. In particular, following the concept of Prager and Synge [64], cf. also Synge [72], Aubin and Burchard [13], and Hlaváček et al. [50], and invoking fluxes in the H(div, Ω) space, guaranteed upper bounds on the error can be obtained. A general functional framework delivering guaranteed upper bounds, independent of the numerical method, has been derived by Repin [66,67,68]. It does not rely on Galerkin orthogonality neither on local equilibration and accommodates an arbitrary flux reconstruction. The idea of using a local residual equilibration procedure for the normal face fluxes reconstruction has been proposed by Ladevèze [55], Ladevèze and Leguillon [56], Kelly [51], Ainsworth and Oden [7,8], and Parés et al. [61,62]. In this context, guaranteed upper bounds typically require solving infinite-dimensional element problems, which, in practice, are approximated. On the other hand, an essential property achieved by means of local equilibration procedures is local efficiency, meaning that the derived estimators also represent local lower bounds of the error, up to a generic constant. This appears to be crucial in view of local mesh refinement, as well as in order to obtain robustness in singularly perturbed problems. Cheap local flux equilibrations leading to a fully computable guaranteed upper bound have been obtained by Destuynder and Métivet [38]. Later, mixed finite element solutions of local Neumann problems posed over patches of (sub)elements, where one minimizes locally the estimator contributions, were proposed, see Luce and Wohlmuth [57], Braess and Schöberl [19],and [77,30,79]. As a matter of fact, lifting the normal face fluxes of the equilibrated residual method to H(div, Ω) immediately yields equilibrated fluxes, cf. Nicaise et al. [59]. Then, both a guaranteed bound and local efficiency are obtained. For computational comparisons of some of these approaches in the lowest-order case, see Carstensen and Merdon [28].The theory in the nonconforming setting, where the discrete solution (potential) is not in the energy space H 1 (Ω), appears to be less developed....

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“…The nonconforming finite element method for (2.2), cf. Stoyan and Baran [71] or [9], reads: find u h ∈ V h such that…”

Section: Nonconforming Finite Elementsmentioning

confidence: 99%

Section: Potential Reconstruction For the Nonsymmetric And Incompletementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Ern¹,

Vohralı́k²

2015

SIAM J. Numer. Anal.

163

196

View full text Add to dashboard Cite

show abstract

“…In some other papers ( [6], [3], [2]) interesting approaches and ideas are given, however only for a posteriori error indicators with respect to the discrete H 1 -seminorm.…”

Section: For a Nonconforming Fem We Havementioning

confidence: 99%

“…In [3] also a class of nonconforming FEM is considered, however only based on a triangular partition and on P l -elements. All these FEM satisfy our assumptions (A1) from Subsection 2.1 and (A2) from Section 3 with k = l − 1.…”

Section: For a Nonconforming Fem We Havementioning

confidence: 99%

New results concerning the DWR method for some nonconforming FEM

Vanselow

2012

Appl Math

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Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity

Ainsworth

Rankin

2010

Numerical Meth Engineering

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We obtain fully computable a posteriori error estimators for the energy norm of the error in second-order conforming and nonconforming finite element approximations in planar elasticity. These estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error. The estimators are shown to also provide local lower bounds, up to a constant and higher-order data oscillation terms. Numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator

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Fully Computable Bounds for the Error in Nonconforming Finite Element Approximations of Arbitrary Order on Triangular Elements

Cited by 30 publications

References 15 publications

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

New results concerning the DWR method for some nonconforming FEM

Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity

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