Discontinuous Galerkin finite element approximation of the two‐dimensional Navier–Stokes equations in stream‐function formulation

Mozolevski, Igor; Süli, Endre; Bösing, Paulo Rafael

doi:10.1002/cnm.944

Cited by 16 publications

(11 citation statements)

References 27 publications

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“…This second IPM method exactly coincides with the DG method proposed in [13], where different alternatives for the approximation, based on the definition of a piecewise continuous stream function spaces, are also proposed and analyzed. In fact, it is worth noting the contributions in solid mechanics by Hansbo and co-workers [14,15,12], which have inspired several authors (see for instance [16] for the solution of the Navier-Stokes equations) and in particular this paper.…”

Section: Introductionmentioning

confidence: 78%

Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations

Montlaur

Fernández‐Méndez

Huerta

2008

Numerical Methods in Fluids

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SUMMARYA Discontinuous Galerkin (DG) method with solenoidal approximation for the simulation of incompressible flow is proposed. It is applied to the solution of the Stokes equations. The Interior Penalty Method is employed to construct the DG weak form. For every element, the approximation space for the velocity field is decomposed as direct sum of a solenoidal space and an irrotational space. This allows to split the DG weak form in two uncoupled problems: the first one solves for the velocity and the hybrid pressure (pressure along the mesh edges) and the second one allows the computation of the pressure in the element interior. Furthermore, the introduction of an extra penalty term leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Numerical examples demonstrate the applicability of the proposed methodologies.

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Section: Introductionmentioning

confidence: 78%

Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations

Montlaur

Fernández‐Méndez

Huerta

2008

Numerical Methods in Fluids

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“…In the present work, we consider the steady incompressible Navier-Stokes equations as a model problem. Various DG approximations of this problem have been investigated recently [3,10,26,27,31]. Here, we identify a set of design conditions on the discrete convective trilinear form to prove convergence.…”

Section: Introductionmentioning

confidence: 99%

Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations

2010

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Abstract. Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier-Stokes equations. Two discrete convective trilinear forms are proposed, a nonconservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.

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“…For more recent developments, see, for example, [54,37,3,15] and references therein. The papers of Babuška and Zlámal [4] and Baker [6] are the earliest contributions to the theory of discontinuous Galerkin finite element methods for fourth-order elliptic problems; for more recent results, including historical notes, see [47,34,49,50,48]. The application of discontinuous Galerkin methods to the Cahn-Hilliard equation is discussed in [20,56,58,35].…”

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confidence: 99%

Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

Kay¹,

Styles²,

Süli³

2009

SIAM J. Numer. Anal.

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Abstract. The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn-Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p ≥ 1 and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken L ∞ (H 1 ) norm with optimal order, O(h p + τ ); the associated chemical potential w = Φ (c) − γ 2 ∆c is shown to be approximated with optimal order, O(h p + τ ), in the broken L 2 (H 1 ) norm. Here Φ(c) = 1 4(1 − c 2 ) 2 is a quartic free-energy function and γ > 0 is an interface parameter. Numerical results are presented with polynomials of degree p = 1, 2, 3.

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Discontinuous Galerkin finite element approximation of the two‐dimensional Navier–Stokes equations in stream‐function formulation

Cited by 16 publications

References 27 publications

Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations

Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations

Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations

Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

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