A discontinuous Galerkin method    with nonoverlapping domain decomposition   for the Stokes and Navier-Stokes problems

Girault, Vivette; Rivière, Béatrice; Wheeler, Mary F.

doi:10.1090/s0025-5718-04-01652-7

Cited by 191 publications

(192 citation statements)

References 28 publications

Supporting

Mentioning

191

Contrasting

Order By: Relevance

“…• The DG methods have been successfully applied to a wide variety of problems ranging from the solid mechanics to the fluid mechanics (see, e.g., [3,7,14,15,17,20,22,40] 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.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Epshteyn¹,

Kurganov²

2009

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Finally the nonlinear convection term u · ∇u is approximated by the following variant of Lesaint-Raviart upwinding (see [21]) that was introduced in [14]. In theory, it is difficult to prove that it brings an improvement, because the Navier-Stokes equation is not purely a transport equation, but in practice, upwinding is useful when the convection is dominant.…”

Section: Discontinuous Galerkin For Both Stepsmentioning

confidence: 99%

“…As far as the inf-sup condition is concerned, it is proven in [14] that the pair of spaces defined by (1.10), (1.11) satisfies a uniform discrete inf-sup condition. More precisely, with the spacē…”

Section: Approximation With Sipgmentioning

confidence: 99%

“…The idea of using a non-symmetric form without penalty was introduced by Baumann and Oden [3]. The NIPG and SIPG methods for the steady-state Navier-Stokes equations were first formulated and analyzed in Girault et al [14]. In Kaya and Rivière [20] both NIPG and SIPG methods coupled with a subgrid eddy viscosity method are applied to the time-dependent NavierStokes problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations

Girault¹,

Rivière²,

Wheeler³

2005

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we solve the time-dependent incompressible Navier-Stokes equations by splitting the non-linearity and incompressibility, and using discontinuous or continuous finite element methods in space. We prove optimal error estimates for the velocity and suboptimal estimates for the pressure. We present some numerical experiments.Mathematics Subject Classification. 65M12, 65M15, 65M60.

show abstract

“…They are based on either overlapping [24,35] or non-overlapping Schwarz methods [1,9,19,25,30,33]. For instance, the Stokes problem was dealt with by a Robin-Robin method in Otto et al [28,32], Chacón Rebollo and Chacón Vera [11,12] and Discacciati et al [15].…”

Section: Introductionmentioning

confidence: 99%

A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem

Benhassine

Bendali²

2010

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at the cross points, that is, points shared by more than two subdomains, one can interpret them as iterative solvers of the actual discrete problem directly issued from the finite element scheme. In this way, the good stability properties of continuous-pressure mixed finite element approximations of the Stokes system are preserved. Estimates ensuring that each iteration can be performed in a stable way as well as a proof of the convergence of the iterative process provide a theoretical background for the application of the related solving procedure. Finally some numerical experiments are given to demonstrate the effectiveness of the approach, and particularly to compare its efficiency with an adaptation to this framework of a standard FETI-DP method.Mathematics Subject Classification. 76D07, 65N55, 65N30.

show abstract

A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems

Cited by 191 publications

References 28 publications

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

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

A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations

A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem

Contact Info

Product

Resources

About