Postprocessing schemes for some mixed finite elements

Stenberg, Rolf

doi:10.1051/m2an/1991250101511

Cited by 135 publications

(143 citation statements)

References 17 publications

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“…Similar post-processing results were obtained for the so-called Brezzi-Douglas-Marini (BDM) mixed method in [9]. Variations on post-processing have been proposed in [8,33,46,47] and an extension to the Hellan-Herrmann-Johnson method in [28].…”

Section: Introductionsupporting

confidence: 55%

To CG or to HDG: A Comparative Study

2011

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Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle hp-adaptivity.

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Section: Introductionsupporting

confidence: 55%

To CG or to HDG: A Comparative Study

2011

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“…Our choice is the postprocessing scheme due to Stenberg [19] which is general enough to cover all existing mixed finite elements, and moreover, does not require the use of Lagrange multipliers.…”

Section: Mixed Finite Element Methods Given a Pair Of Finite Elementmentioning

confidence: 99%

A posteriori error analysis for locally conservative mixed methods

Kim

2007

Math. Comp.

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Abstract. In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart-Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the P 1 nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.

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“…Different postprocessing techniques for mixed finite elements have been introduced in the past. Let us cite the works of Arnold and Brezzi [9], Bramble and Xu [16], Stenberg [57], Chen [25], Arbogast and Chen [8], and, for the lowest-order Raviart-Thomas-Nédélec case, the author [62]. It will turn out that for our purposes, the postprocessing of [62] and [8] under Assumption (A) and that of [8] in general will be optimal.…”

Section: 2mentioning

confidence: 99%

“…Postprocessing of p h into a new approximationp h is then usually used for the double purpose of giving an improved approximation to p and facilitating the implementation of mixed methods; cf. Arnold and Brezzi [9], Bramble and Xu [16], Stenberg [57], Chen [25], and Arbogast and Chen [8]. In combination with mesh-dependent norms, it has also previously been used in order to obtain error estimates in, e.g., Lovadina and Stenberg [44]; see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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Postprocessing schemes for some mixed finite elements

Cited by 135 publications

References 17 publications

To CG or to HDG: A Comparative Study

To CG or to HDG: A Comparative Study

A posteriori error analysis for locally conservative mixed methods

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

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