The local discontinuous Galerkin method for the Oseen equations

Cockburn, Bernardo; Kanschat, Guido; Schötzau, Dominik

doi:10.1090/s0025-5718-03-01552-7

Cited by 98 publications

(86 citation statements)

References 30 publications

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“…The weak discontinuous Galerkin elemental formulation (15) and (16) (20) where if = 0, Equation (19) reduces to Equation (15). A similar modification can also be applied to Equation (17).…”

Section: Elemental Formulationmentioning

confidence: 99%

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On 2D elliptic discontinuous Galerkin methods

Sherwin

Kirby

Peiró

et al. 2005

Numerical Meth Engineering

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SUMMARYWe discuss the discretization using discontinuous Galerkin (DG) formulation of an elliptic Poisson problem. Two commonly used DG schemes are investigated: the original average flux proposed by Bassi and Rebay (J. Comput. Phys. 1997 1119 -1148) by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the DG method. Through consideration of standard C 0 -type expansion bases, as opposed to elementally orthogonal expansions, with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted. The use of C 0 -type expansions also permits the direct enforcement of Dirichlet boundary conditions through a 'lifting' approach where the LDG flux does not require further stabilization. In our construction we also adopt a formulation of the continuous DG fluxes that permits a more general interpretation of their numerical implementation. In particular it allows us to determine the conditions under which the LDG method provides a near local stencil. Finally a study of the conditioning and the size of the null space of the matrix systems resulting from the DG discretization of the elliptic problem is undertaken.

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“…The weak discontinuous Galerkin elemental formulation (15) and (16) (20) where if = 0, Equation (19) reduces to Equation (15). A similar modification can also be applied to Equation (17).…”

Section: Elemental Formulationmentioning

confidence: 99%

“…Specific details regarding the theoretical foundations for the multi-dimensional LDG have been presented in References [14,17,18]. We also note that the LDG has been successfully applied in the solution of non-trivial elliptic problems such as the Stokes system [19] and the Oseen equations [20].…”

Section: Introductionmentioning

confidence: 99%

On 2D elliptic discontinuous Galerkin methods

Sherwin

Kirby

Peiró

et al. 2005

Numerical Meth Engineering

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“…• 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.

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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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“…Para mais detalhes veja [7] e [10]. Em [2], [5] e [4] foi introduzido, analisado e numericamente testado, um método de elementos finitos de Galerkin descontínuo local para as equações de Stokes, equações de Oseen, e para as equações de Navier-Stokes incompressíveis, respectivamente (veja [3] para uma revisão). Nestes trabalhos, foi demonstrado que o métodoé um método misto estabilizado, localmente conservativo e tem alta ordem de precisão.…”

Section: Introductionunclassified

Método de Elementos Finitos de Galerkin Descontínuo para Equações de Navier-Stokes Bidimensionais

Mozolevski

Bösing²

2005

TEMA - Tendências Em Matemática Aplicada E Computacional

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Resumo. Neste trabalho apresentamos um método de Galerkin descontínuo para as equações de Navier-Stokes incompressíveis, bidimensionais em regime permanente. Usando a formulação da função corrente, o problema se reduz para uma equação biharmônica não-linear queé linearizada com o método de iteração de Picard. Para a equação biharmônica linear, apresentamos uma formulação com penalização interior do método de elementos finitos de Galerkin descontínuo. Esta formulaçãoé o resultado da combinação de outras duas formulações, uma para a parte elíptica e outra para parte hiperbólica do problema. São apresentados resultados numéricos que confirmam a eficiência do método na resolução numérica das equações de Navier-Stokes para uma ampla escala do número de Reynolds. IntroduçãoA resolução numérica das equações de Navier-Stokes tem sido o objeto de estudo em dinâmica de fluidos computacionais há muitas décadas. Em particular, os avanços tecnológicos dosúltimos anos contribuíram fortemente para o desenvolvimentos destas pesquisas. Em [8], pode-se encontrar uma revisão dos métodos numéricos mais comuns aplicáveisàs equações de Navier-Stokes incompressíveis. Dentre estes métodos, os métodos de elementos finitos sempre tiveram grande destaque devido a sua adaptatividade, estabilidade e precisão. Para mais detalhes veja [7]

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The local discontinuous Galerkin method for the Oseen equations

Cited by 98 publications

References 30 publications

On 2D elliptic discontinuous Galerkin methods

On 2D elliptic discontinuous Galerkin methods

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

Método de Elementos Finitos de Galerkin Descontínuo para Equações de Navier-Stokes Bidimensionais

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