Mixed Finite Element Methods

Durán, Ricardo G.

doi:10.1007/978-3-540-78319-0_1

Cited by 20 publications

(16 citation statements)

References 47 publications

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“…Let us prove for example the first equality; the other two are obviously analogous. Since div u = 0, we have that div Π k u = 0 and therefore, from a wellknown property of the Raviart-Thomas interpolation (see for example [7,13]), it follows that Π k u ∈ P k ( T ) 3 . On the other hand, now using (2.1) for i = 2, 3, and that u 2 = u 3 = 0, we have…”

Section: Their Raviart-thomas Interpolations Are Of the Same Form; Namentioning

confidence: 99%

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Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

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Abstract. We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra.Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k ≥ 0, we prove error estimates of order j + 1 when the vector field being approximated has components in W j+1,p , for triangles or tetrahedra, where 0 ≤ j ≤ k and 1 ≤ p ≤ ∞. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.

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Section: Their Raviart-thomas Interpolations Are Of the Same Form; Namentioning

confidence: 99%

“…In particular the inf-sup condition can be obtained from this estimate (see for example [11,13]). The maximum angle condition was originally introduced for triangles.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

et al. 2010

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“…We are going to use the well known theory developed by Brezzi (see for example [9,15,26]) but modifying the usual Hilbert spaces and the bilinear form corresponding to the divergence free restriction in the weak formulation of the Stokes equations. …”

Section: An Application To the Stokes Equationsmentioning

confidence: 99%

SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR Hölder-Α DOMAINS

Durán

García

2010

Math. Models Methods Appl. Sci.

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Abstract.If Ω ⊂ R n is a bounded domain, the existence of solutions u ∈ H 1 0 (Ω) n of div u = f for f ∈ L 2 (Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution (u,, where u is the velocity and p the pressure. It is known that the above mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains.In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal.For some particular domains with an external cusp we apply our results to show the well posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u, p) ∈ H 1 0 (Ω) n × L r (Ω) for some r < 2 depending on the power of the cusp.

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“…• (see [7,12]), the latter being a closed subspace of (V 0 ) ′ . Finally, we define the linear and continuous operator πA :…”

Section: Well-posedness and Euler-lagrange Equationsmentioning

confidence: 99%

Extended Variational Formulation for Heterogeneous Partial Differential Equations

Blanco

Gervasio

Quarteroni

2011

Comput. Methods Appl. Math.

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-We address the coupling of an advection equation with a diffusionadvection equation, for solutions featuring boundary layers. We consider nonoverlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out.2010 Mathematical subject classification: 35Q35; 65N55; 65N35; 65F08.

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Mixed Finite Element Methods

Cited by 20 publications

References 47 publications

Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR Hölder-Α DOMAINS

Extended Variational Formulation for Heterogeneous Partial Differential Equations

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