Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods

Douglas, Jim; Dupont, Todd

doi:10.1007/bfb0120591

Cited by 395 publications

(353 citation statements)

References 2 publications

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“…In particular, a finite difference-method of characteristics scheme has been considered [5] by one of the authors ; the analysis of that combination is based in part on the results of this paper. It would also be possible to use a finite element-method of characteristics procedure, as was discussed in the thesis of T. F. Russell [11], or an interior penalty Galerkin method [1,3,4,14] ; however, since the main point of this paper is to show the feasibility of the use of the mixed method for the pressure, we shall confine our treatment of the concentration to the single case.…”

Section: Simulation Of Miscible Displacement 19mentioning

confidence: 99%

The approximation of the pressure by a mixed method in the simulation of miscible displacement

Douglas¹,

Ewing²,

Wheeler³

1983

RAIRO. Anal. numér.

303

225

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Section: Simulation Of Miscible Displacement 19mentioning

confidence: 99%

The approximation of the pressure by a mixed method in the simulation of miscible displacement

Douglas¹,

Ewing²,

Wheeler³

1983

RAIRO. Anal. numér.

303

225

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“…In this paper, we develop a family of high-order DG methods for the system (1.8). The proposed methods are based on three primal DG methods: the Nonsymmetric Interior Penalty Galerkin (NIPG), the Symmetric Interior Penalty Galerkin (SIPG), and the Incomplete Interior Penalty Galerkin (IIPG) methods, [4,18,19,39]. The numerical fluxes in the proposed DG methods are the fluxes developed for the semidiscrete finite-volume central-upwind schemes in [32] (see also [31,33] and references therein).…”

Section: Introductionmentioning

confidence: 99%

New Interior Penalty Discontinuous Galerkin Methods for the Keller–Segel Chemotaxis Model

Epshteyn¹,

Kurganov²

2009

SIAM J. Numer. Anal.

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We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolic-elliptic type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusion-reaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws.In this paper, we consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. We also show that the blow-up time of the exact solution is bounded from above by the blow-up time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillation-free, even though no slope limiting technique has been implemented.

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“…The Continuous Interior Penalty (CIP) method was originally proposed by Douglas and Dupont [9] for parabolic and elliptic equations. The idea was to add a penalization term on the gradient jumps in order to increase robustness for elliptic problems with a dominating convection term.…”

Section: Introductionmentioning

confidence: 99%

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman

Guzmán

Leykekhman

2008

IMA Journal of Numerical Analysis

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Abstract.In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convection-dominated singularly perturbed convection-diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods.

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Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods

Cited by 395 publications

References 2 publications

The approximation of the pressure by a mixed method in the simulation of miscible displacement

The approximation of the pressure by a mixed method in the simulation of miscible displacement

New Interior Penalty Discontinuous Galerkin Methods for the Keller–Segel Chemotaxis Model

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

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