On the Two-Dimensional Navier–Stokes Equations in Stream Function Form

Guo, Bing; He, Lu; De-kang, Mao

doi:10.1006/jmaa.1996.5174

Cited by 9 publications

(5 citation statements)

References 2 publications

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“…The idea of the HHJ formulation is, similarly as in the derivation of the MCS formulation (6), motivated by rewriting the fourth order problem of the stream function formulation (7) as a mixed system. To his end we introduce the space…”

Section: A Hellan-herrmann-johnson Like Methods For the Stream Functi...mentioning

confidence: 99%

“…As (6c) gives div(u MCS ) = 0, it follows that the pair (σ MCS , u MCS ) is uniquely defined by testing equation (6b) only with divergence free test functions v ∈ V 0 := {v ∈ V : div(v) = 0}. Since the pair (σ MCS , u) is also a solution of (6b) (on V 0 ), we have that u = u MCS by the uniqueness of the solution of equation (6).…”

Section: A Weak Formulation With Reduced Regularitymentioning

confidence: 97%

“…However, we want to emphasize that the methods proposed in this work are of optimal order. In [6,5,4], the authors focused on the pure stream function formulation (in two space dimensions) given by (2) and derived a finite difference scheme for the approximation of the bi-Laplacian operator. A mixed finite element method was derived in [13].…”

Section: Introductionmentioning

confidence: 99%

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A Hellan-Herrmann-Johnson-like method for the stream function formulation of the Stokes equations in two and three space dimensions

Lederer¹

2020

Preprint

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We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan-Herrmann-Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P.L. Lederer, J. Schöberl, IMA Journal of numerical Analysis, 2019] that approximates the velocity in an H(div)-conforming space and introduces a new stress-like variable for the approximation of the gradient of the velocity within the function space H(curl div). The properties of the (discrete) de Rham complex allows to extend this method to a stream function formulation in two and three space dimensions. We present a detailed stability analysis in the continuous and the discrete setting where the stream function ψ and its approximation ψ h are elements of H(curl) and the H(curl)-conforming Nédélec finite element space, respectively. We conclude with an error analysis revealing optimal convergence rates for the error of the discrete velocity u h = curl(ψ h ) measured in a discrete H 1 -norm. We present numerical examples to validate our findings and discuss structure-preserving properties such as pressure-robustness.Stokes equations, Hellan-Herrmann-Johnson, Stream function formulation, incompressible flows.represent the vector valued and matrix-valued versions of). This notation is extended in an obvious fashion to other function spaces as needed.Depending on the type of the function, the gradient ∇ is to be understood from the context as an operator that results in either a vector whose components are [∇φ] i = ∂ i φ (where ∂ i is the partial derivative ∂/∂x i ) for φ ∈ C ∞ c ′ (Ω, R) or a matrix whose entries are [∇φ] ij = ∂ j φ i for φ ∈ C ∞ c ′ (Ω, R d ). Similarly, the "curl" is given as any of the following

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Section: A Hellan-herrmann-johnson Like Methods For the Stream Functi...mentioning

confidence: 99%

Section: A Weak Formulation With Reduced Regularitymentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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A Hellan-Herrmann-Johnson-like method for the stream function formulation of the Stokes equations in two and three space dimensions

Lederer¹

2020

Preprint

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“…Recently the authors considered this problem. The existence, uniqueness and regularity of the weak solution are proved under some conditions in [16]. While a fully discrete Legendre spectral scheme is proposed in [17].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we introduce the weak formulation of (1.1) and some results in [16,17]. In Section 3, we construct the predictioncorrection scheme.…”

Section: Introductionmentioning

confidence: 99%

Prediction-correction legendre spectral scheme for incompressible fluid flow

He¹,

De-kang²,

Guo³

1999

ESAIM: M2AN

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Abstract. The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A prediction-correction Legendre spectral scheme is proposed, which is easy to be performed. The numerical solution possesses the accuracy of second-order in time and higher order in space. The numerical experiments show the high accuracy of this approach.Résumé. Le problème de fluide incompressibleà deux dimensions est considéré sous forme de fonctioncourant. Un schéma de type prédiction-correction spectrale de Legendre est proposé, ce dernierétant facileà mettre en oeuvre. La solution numérique possède une précision de second ordre en temps et d'ordre supérieur en espace. Les résultats numériques montrent la grande préscison de cette approche.

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