Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux

Pietro, Daniele Antonio Di

doi:10.1002/fld.1495

Cited by 12 publications

(10 citation statements)

References 17 publications

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“…Observe that the above argument does not entail restrictions on the polynomial degree k in (12). This is so because the pressure is sought in a continuous space.…”

Section: Space Discretizationmentioning

confidence: 99%

“…We refer to Toselli [36] for a comprehensive study of fully discontinuous space couples on quadrilateral and hexahedral meshes. Stabilized formulations allowing equal-order, fully discontinuous approximations can be found, e.g., in [4,5,10,8,9,12,13].…”

Section: Space Discretizationmentioning

confidence: 99%

“…More general meshes and equal-order approximation can be dealt with by suitable pressure stabilization techniques. We refer, in particular, to Cockburn Kanschat, Schötzau and Schwab [9] and to Bassi, Crivellini, Di Pietro and Rebay [4,5]; see also [12]. Several techniques have also been proposed for the discretization of the non-linear convective term.…”

Section: Introductionmentioning

confidence: 99%

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A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure

Botti

Pietro

2011

Journal of Computational Physics

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In this work we present a pressure-correction scheme for the incompressible Navier-Stokes equations combining a discontinuous Galerkin approximation for the velocity and a standard continuous Galerkin approximation for the pressure. The main interest of pressure-correction algorithms is the reduced computational cost compared to monolithic strategies. In this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high Reynolds regimes. The proposed spatial velocity-pressure approximation is LBB stable for equal polynomial orders and it allows adaptive p-refinement for velocity and global p-refinement for pressure. The method is validated against a large set of classical two-and three-dimensional test cases covering a wide range of Reynolds numbers, in which it proves effective both in terms of accuracy and computational cost.

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“…Observe that the above argument does not entail restrictions on the polynomial degree k in (12). This is so because the pressure is sought in a continuous space.…”

Section: Space Discretizationmentioning

confidence: 99%

Section: Space Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure

Botti

Pietro

2011

Journal of Computational Physics

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“…The velocity-pressure coupling hinges on the bilinear form b h ∈ L(U h × P h , R) (see, e.g., [12]):…”

Section: Remark 23 (Numerical Integration)mentioning

confidence: 99%

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

2012

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Abstract. In this work we propose an original implementation of a large family of lowest-order methods for diffusive problems including standard and hybrid finite volume methods, mimetic finite difference-type schemes, and cell centered Galerkin methods. The key idea is to regard the method at hand as a (Petrov-)Galerkin scheme based on possibly incomplete, broken affine spaces defined from a gradient reconstruction and a point value. The resulting unified framework serves as a basis for the development of a FreeFEM-like domain specific language targeted at defining discrete linear and bilinear forms. Both the back-end and the front-end of the language are extensively discussed, and several examples of applications are provided. The overhead of the language is evaluated by comparing with a more traditional implementation. A benchmark including the comparison with more classical finite element methods on standard meshes is also proposed.Key words. Domain specific embedded language, finite volume methods, cell centered Galerkin methods, Petrov-Galerkin methods AMS subject classifications. 65N08, 65N30, 65Y051. Introduction. Lowest-order methods possibly featuring conservation of physical quantities are traditionally employed in industrial applications where computational cost is a crucial issue. In this context, the use of general polyhedral, possibly nonconforming meshes commends itself for a number of reasons. To cite a few: (i) remeshing can be avoided or postponed in problems that involve mesh deformation -e.g. in sedimentary basin modeling non-standard elements and nonconformities can appear due to the erosion of geological layers; -(ii) the number of degrees of freedom can be reduced by aggregative coarsening techniques -cf. [7] for an application in the context of discontinuous Galerkin (dG) methods; -(iii) geometrical features can be represented more accurately without unduly increasing the number of mesh elements.Handling general polyhedral meshes requires numerical schemes that possess the usual properties of stability and consistency. In the context of cell centered finite volume methods, a popular way to achieve consistency on general polyhedral meshes is provided by Multipoint Finite Volume schemes independently introduced by Aavatsmark, Barkve, Bøe and Mannseth [2] and Edwards and Rogers [22]. The main advantage of multipoint schemes is that they can be easily fitted into existing simulators based on standard finite volume schemes. A major drawback is their lack of stability in some configurations. Two ways of overcoming this difficulty by designing discretizations based on the variational formulation of the problem and featuring cell-and face-unknowns have been proposed by Brezzi, Lipnikov and coworkers [8,9] (Mimetic Finite Difference methods) and by Droniou and Eymard [20] (Mixed/Hybrid Finite Volume methods). In this context, Eymard, Gallouët and Herbin [24] have shown that face unknowns can be selectively used as Lagrange multipliers to enforce flux continuity, or eliminated using a consisten...

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“…In our case, the velocity-pressure coupling is stabilized by penalizing the pressure jumps across interfaces with a weight proportional to the meshsize; see, e.g., [19]. As regards the convective term, we use the non-dissipative trilinear form recently proposed by Di Pietro and Ern [22], which has proven suitable to convection-dominated regimes; see also Botti and Di Pietro [9] for the application to a dG discretization of the advection step in the context of a pressure-correction time-integration scheme.…”

Section: The Discrete Settingmentioning

confidence: 99%

Cell centered Galerkin methods for diffusive problems

Pietro¹

2011

ESAIM: M2AN

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Abstract. In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.

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Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux

Cited by 12 publications

References 17 publications

A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure

A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

Cell centered Galerkin methods for diffusive problems

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