Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

Roos, Hans–Görg

doi:10.5402/2012/379547

Cited by 22 publications

(13 citation statements)

References 130 publications

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“…Due to Lemma 3.3 and noting p · n K = p h · n K on each edge of K, one can obtain a fully computable local error estimatorη 20) and the corresponding global oneη…”

Section: )mentioning

confidence: 99%

“…Even for a priori error analysis on this problem, we only find some nonconforming method in [27]. For completeness, we refer the readers to the review article [20] by Roos. In this paper, we derive guaranteed a posteriori error estimates for the nonconforming CrouzeixRaviart finite element approximation [9] of problem (1.1). In contrast to the conforming case, there appears the nonconforming term in error analysis.…”

Section: Introductionmentioning

confidence: 99%

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Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

Zhang

Chen

Zhao

2015

Applied Numerical Mathematics

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“…Due to Lemma 3.3 and noting p · n K = p h · n K on each edge of K, one can obtain a fully computable local error estimatorη 20) and the corresponding global oneη…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

Zhang

Chen

Zhao

2015

Applied Numerical Mathematics

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“…It is also well-known [25,27] that solutions to the problem (1.1) may develop layers, whose approximation is the major difficulty of designing high-order, robust numerical schemes. We refer to [39,38] for a comprehensive information on different numerical techniques for (1.1).…”

Section: Introductionmentioning

confidence: 99%

An analysis of HDG methods for convection-dominated diffusion problems

Qiu

Zhang

2015

ESAIM: M2AN

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Abstract. We present the first a priori error analysis of the h-version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection-dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L 2 -error of the scalar variable converges with order k + 1/2 on general conforming quasi-uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L 2 -convergence order of k + 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results.

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“…However, the numerical approximation of those equations is a challenging problem, in general, and it is not possible to get reasonable approximations with the standard numerical algorithms. Therefore developing efficient and effective computational methods for solving those problems has been drawing attention of many researchers for several decades [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems

Sendur

2018

Journal of Applied Mathematics

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The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convectiondominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations.

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Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

Cited by 22 publications

References 130 publications

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

An analysis of HDG methods for convection-dominated diffusion problems

A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems

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