Analysis of mixed methods using mesh dependent norms

Babuška, Ivo; Osborn, John E.; Pitkäranta, Juhani

doi:10.1090/s0025-5718-1980-0583486-7

Cited by 185 publications

(143 citation statements)

References 12 publications

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“…The variational formulation of (3.3), upon which the finite element method is based, can be stated in different ways ; cf. [3,8], They ail, however, lead to the same discretization and hence we will turn directly to that. For the index k, k^l, and for a regular triangular paritioning 1S A , the finite element spaces are defîned through For the error analysis of the method we refer directly to the papers [3] and [8].…”

Section: ôVmentioning

confidence: 99%

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Postprocessing schemes for some mixed finite elements

Stenberg¹

1991

ESAIM: M2AN

135

134

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Section: ôVmentioning

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%

Postprocessing schemes for some mixed finite elements

Stenberg¹

1991

ESAIM: M2AN

135

134

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“…Traditionally, the natural norms of the spaces H(div, Ω) and L 2 (Ω) are used, but mesh-dependent norms can be employed instead; cf. Babuška et al [10]. Postprocessing of p h into a new approximationp h is then usually used for the double purpose of giving an improved approximation to p and facilitating the implementation of mixed methods; cf.…”

Section: Introductionmentioning

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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“…Indeed, appropriate mesh dependent norms on X h × M h , denoted ω 0,h and φ 2,h (see [2] for the definitions) need to be introduced if the natural approximation spaces of X h consisting of C 0 piecewise polynomials of degree k ≥ 1 and M h = X h ∩ H 1 0 (Ω), are to satisfy Brezzi's abstract stability conditions, see [5]. This intrinsic lack of stability is what ultimately limits the potential for developing spectrally equivalent preconditioners (and robust multigrid methods) for solving the discretised problem (1.3).…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, infsup stability and boundedness with respect to the mesh dependent norms · 2,h and · 0,h , see [2,Thm. 3], implies that…”

Section: Introductionmentioning

confidence: 99%

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Silvester

Mihajlović

2004

BIT Numerical Mathematics

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Abstract.We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. (2000): 65F10, 65N12, 65N22, 65N55. AMS subject classification

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Analysis of mixed methods using mesh dependent norms

Cited by 185 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 Black-Box Multigrid Preconditioner for the Biharmonic Equation

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