Efficient rectangular mixed finite elements in two and three space variables

Brezzi, Franco; Douglas, Jim; Fortin, Michel; Marini, L. D.

doi:10.1051/m2an/1987210405811

Cited by 286 publications

(245 citation statements)

References 11 publications

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“…The analysis of a special finite element approximation of the scheme (2 2) was simphfied by the existence of a discrete version of the Helrnholtz Theorem [1] Although a décomposition is not always given m this strong sensé, a certain step in this direction was done when Brezzi et al [5] introduced the following five properties Assume that besides [13], whereas P 4 states that the pair (F^9 Q h ) is « good » for the mixed variable formulation of an elliptic second order équation [6,7,11,15] Brezzi, Bathe and Fortin [5] used these properties for the analysis of the plate in the limit case t = 0 It is our aim to show that the properties P x to F 5 also imply convergence which is uniformly good for every t > 0 To this end we will extend the Helmholtz décomposition to the finite element spaces…”

Section: (O) and || || S Is The Norm In The Sobolev Spaces H S {O) mentioning

confidence: 99%

Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates

Peisker¹,

Braess²

1992

ESAIM: M2AN

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Section: (O) and || || S Is The Norm In The Sobolev Spaces H S {O) mentioning

confidence: 99%

Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates

Peisker¹,

Braess²

1992

ESAIM: M2AN

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“…be any of the usual mixed finite element spaces (e.g., those of [9,10,30,11,37]), and let V h or, equivalently, V h Á m contain the polynomials of degree k. Then let:…”

Section: The Finite Element Approximationmentioning

confidence: 99%

A multiscale preconditioner for stochastic mortar mixed finite elements

Wheeler

Wildey

Yotov

2011

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThe aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy's law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.

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“…Other families of spaces, in both 2 and 3 dimensions which are intermediate between the RT and the BDM spaces were introduced and analyzed by Brezzi, Douglas, Fortin and Marini [15].…”

Section: Theorem 310 If the Solution (U P) Of Problem 32 Belongs Tomentioning

confidence: 99%