An accurate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi mathvariant="bold">H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">div</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> flux reconstruction for discontinuous Galerkin approximations of elliptic problems

Ern, Alexandre; Nicaise, Serge; Vohralı́k, Martin

doi:10.1016/j.crma.2007.10.036

Cited by 78 publications

(86 citation statements)

References 7 publications

(12 reference statements)

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“…Let p ∈ H m+1 (Ω) be the solution to the dual problem (7), with m > k, and suppose that the Riesz representer q of the goal functional has oscillations of order m − 1 in V m−1 h . Let u h ∈ V k h and p h ∈ V m h be the dG finite element solutions to (26) and (27), respectively. Then, there exists a constant C, depending only on the shape regularity of T h , the data {D, f, q} in the primal and dual problems, and the exact solutions {u, p}, such that…”

Section: Goal-oriented Error Estimates For Dg Methodsmentioning

confidence: 99%

“…Furthermore, the dG method benefits from the fact that it is locally conservative; it implies that the construction of equilibrated fluxes, needed for the evaluation of the proposed error estimator, is rather straightforward, see e.g. [36,26,19,12]. Finally, we show that, depending on the approximation properties of the primal method and problem data, the order of the dG method, used to solve the dual problem, can be chosen in such a way that the respective error estimator is asymptotically exact.…”

Section: Introductionmentioning

confidence: 99%

“…Advantages of the dG method are that high-order elements in any space dimension can be easily implemented and that it is locally conservative, which yields in a straightforward manner equilibrated fluxes. In fact, it has been shown to provide cheap and local reconstruction algorithms [35,26,19,12].…”

mentioning

confidence: 99%

“…In fact, various flux reconstruction techniques have been developed over the years in the case of elliptic problems and some of the techniques have been extended to non-conforming conservative methods [36,26,12], to first-order conforming finite element method [13], or to the finite volume and finite difference methods [54].…”

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confidence: 99%

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Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Mozolevski

Prudhomme

2015

Computer Methods in Applied Mechanics and Engineering

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We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dualweighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed RaviartThomas method are used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

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Section: Goal-oriented Error Estimates For Dg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Mozolevski

Prudhomme

2015

Computer Methods in Applied Mechanics and Engineering

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“…On the other hand, the discontinuous Galerkin method becomes recently very popular and is a very efficient tool for the numerical approximation of reaction-convection-diffusion problems for instance. Some a posteriori error analysis were performed recently, let us quote [1,2,7,10,14,16,18,20,21,26] for pure diffusion (or diffusion-dominated) problems and [9,13,16] for singularly perturbed problems (i.e., dominant advection or reaction); for Maxwell system see for instance [17]. In all these papers, no convergence results are proved and to our knowledge, only the recent paper of Karakashian and Pascal [19] provides a convergence result for a purely diffusion problem.…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Nicaise

Cochez-Dhondt

2010

ESAIM: M2AN

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Abstract. We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V . We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convectionreaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.Mathematics Subject Classification. 65N30, 65N15, 65N50.

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Discontinuous Galerkin Methods for Computational Fluid Dynamics

Cockburn

2017

Encyclopedia of Computational Mechanics Second Edition

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The discontinuous Galerkin (DG) methods are locally conservative, high‐order accurate, robust methods that can easily handle elements of arbitrary shapes, irregular meshes with hanging nodes, and polynomial approximations of different degrees in different elements. These properties, which render them ideal for hp ‐adaptivity in domains of complex geometry, have brought them to the main stream of computational fluid dynamics. We study the properties of the DG methods as applied to a wide variety of problems arising in fluid dynamics with a special emphasis on linear, symmetric positive hyperbolic systems, the Euler equations of gas dynamics, purely elliptic problems, and the incompressible and compressible Navier–Stokes equations. In each instance, we discuss the main properties of the methods, display the mechanisms that make them work so well, and present numerical experiments showing their performance.

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An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems

Cited by 78 publications

References 7 publications

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Discontinuous Galerkin Methods for Computational Fluid Dynamics

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