Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

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“…see [10,20]. A more detailed discussion on the proper choice of the interpolation degree q is given in sections 4.2 and 4.3.…”

Section: Theoremmentioning

confidence: 99%

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Numerical Meth Engineering

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“…may be related to the decomposition of the space H(div, Ω k ) used in the context of mixed or hybrid elements, see [25].…”

Section: Remark 3 Note That the Relaxed Problem (42) Admits At Leasmentioning

confidence: 99%

“…Actually, for ease of explanation, this section provides the strong form of the local error equation (5.3). A more detailed construction of the dual estimates may be found in [25].…”

Section: Strong Form Of the Local Problems For The Dual Estimatesqmentioning

confidence: 99%

Exact Bounds for Linear Outputs of the Advection-Diffusion-Reaction Equation Using Flux-Free Error Estimates

Mariné¹,

Dı́ez

Huerta

2009

SIAM J. Sci. Comput.

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Abstract. The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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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Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

Cited by 33 publications

References 30 publications

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Exact Bounds for Linear Outputs of the Advection-Diffusion-Reaction Equation Using Flux-Free Error Estimates

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

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