Finite Element Methods for Navier-Stokes Equations

Girault, Vivette; Raviart, Pierre-Arnaud

doi:10.1007/978-3-642-61623-5

Cited by 4,190 publications

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“…The equality in part (a) is classical and proved for example in [12,11]. The error estimate is then standard for the interpolation operator in S h (the restriction on δ ensures that p is continuous and hence can be interpolated).…”

Section: Tetrahedral Elementsmentioning

confidence: 99%

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Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Monk

Demkowicz

2000

Math. Comp.

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Abstract. We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

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Section: Tetrahedral Elementsmentioning

confidence: 99%

“…The constraint ν × u h = 0 on Γ is easily implemented by taking the degrees of freedom associated with edges or faces on Γ to be zero [12].…”

Section: Tetrahedral Elementsmentioning

confidence: 99%

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Monk

Demkowicz

2000

Math. Comp.

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“…The space H 0 (curl; Ω) admits the well-known Helmholtz-Hodge decomposition [12,15] for all ψ ∈ H 1 0 (Ω). Since φ can be obtained from the Poisson equation (1.4), we will focus on (1.3), which will be referred to as the reduced time-harmonic Maxwell (RTHM) equations.…”

Section: Introductionmentioning

confidence: 99%

A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations

Brenner

Sung

2006

Math. Comp.

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Abstract. A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free CrouzeixRaviart nonconforming P 1 vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

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“…Assume that (u 1 , 1 ) and (u 2 , 2 ) are two different solutions of (47). From (19) in Lemma 3 we obtain (w, u, u) = 0 ∀w, u ∈ W. Then, we obtain…”

Section: Uniqueness Of Weak Solutionmentioning

confidence: 85%

Solvability of the Brinkman-Forchheimer-Darcy Equation

Skrzypacz

Wei

2017

Journal of Applied Mathematics

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The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy's range.

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Finite Element Methods for Navier-Stokes Equations

Cited by 4,190 publications

References 0 publications

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations

Solvability of the Brinkman-Forchheimer-Darcy Equation

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