Mixed finite elements for second order elliptic problems in three variables

Brezzi, Franco; Douglas, Jim; Durán, Ricardo G.; Fortin, Michel

doi:10.1007/bf01396752

Cited by 328 publications

(287 citation statements)

References 12 publications

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“…The extension to the other mixed methods of [2,4,14,15] is trivial. Hence, we let 'ü h be a regular partitioning of Ù into closed triangles or tetrahedrons and deflne the finite element spaces as [2,4] H" = {peH| p| r G [P k (T) where P{(T), I -k, k -1, / ^ 0, dénotes the polynomials of degree / on T. In [2,4] quasioptimal error estimâtes have been derived for the above method.…”

Section: Second Order Eixïptic Problemsmentioning

confidence: 99%

“…e.g. [7,18] and the références therein) and problems in semiconductor physics [13], and for these two applications very good results have been obtained with the mixed methods of the Raviart-Thomas-Nedelec (RTN) [14,15] and Brezzi-Douglas-Marini (BDM) [2,4] families.…”

Section: Introductionmentioning

confidence: 99%

“…Our exposition will be rather brief, since most of the estimâtes we will need for our analysis are found in [2,3,4,8]. Our notation will be the established one.…”

Section: Introductionmentioning

confidence: 99%

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Postprocessing schemes for some mixed finite elements

Stenberg¹

1991

ESAIM: M2AN

142

134

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Section: Second Order Eixïptic Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Postprocessing schemes for some mixed finite elements

Stenberg¹

1991

ESAIM: M2AN

142

134

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“…Table 1 lists the most common mixed finite element spaces V h (K) × Φ h (K) on an element K ∈ T h . The notation RTN stands for the Raviart-Thomas [53] space on triangles and rectangles and the Nédélec [47] space on tetrahedra and rectangular parallelepipeds if d = 3 and BDM for the Brezzi-Douglas-Marini [19] space on triangles and rectangles and the Brezzi-Douglas-Durán-Fortin [18] space on tetrahedra and rectangular parallelepipeds if d = 3. In the notation, "s" stands for simplices, "r" for rectangular parallelepipeds, P * 2,k := r∇×(x k+1 y)+s∇×(xy k+1 ), r, s ∈ R, and P *…”

Section: The Mixed Finite Element Methodsmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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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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Mixed finite elements for second order elliptic problems in three variables

Cited by 328 publications

References 12 publications

Postprocessing schemes for some mixed finite elements

Postprocessing schemes for some mixed finite elements

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

A multiscale preconditioner for stochastic mortar mixed finite elements

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