<i>A posteriori</i> error estimates for the mixed finite element method with Lagrange multipliers

Gatica, Gabriel N.; Maischak, Matthias

doi:10.1002/num.20050

Cited by 18 publications

(20 citation statements)

References 39 publications

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“…The efficiency of the estimator M ⊕ can be easily evaluated using the lower bound (16). Namely, for the effectivity index of M ⊕ we have:…”

Section: Estimate In the Full Normmentioning

confidence: 99%

“…The residual-based estimates were developed in [2], [6], [12], [1], [16] for the diffusion-type equation and extended in [14] and [20] to the equations of linear elasticity. The superconvergence-based (averaging-type) error estimators were proposed in [7] and [13] to control the L 2 -error of the flux variable.…”

Section: Introductionmentioning

confidence: 99%

“…The superconvergence-based (averaging-type) error estimators were proposed in [7] and [13] to control the L 2 -error of the flux variable. Further, the estimators based on the solution of local problems were presented in [2], [16] and [20], and the hierarchical estimator can be found in [31]. Finally, a comparison of these four types of error estimators for mixed finite element discretizations by Raviart-Thomas elements was presented in [31].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Repin

Sauter²,

Smolianski³

2007

SIAM J. Numer. Anal.

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The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and the individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As the applications, the diffusion problem as well as the problem of linear elasticity are considered.

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“…The efficiency of the estimator M ⊕ can be easily evaluated using the lower bound (16). Namely, for the effectivity index of M ⊕ we have:…”

Section: Estimate In the Full Normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Repin

Sauter²,

Smolianski³

2007

SIAM J. Numer. Anal.

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“…is a generic expression denoting one or several terms of higher order. While the list of references on a-posteriori error analysis for mixed formulations of linear and nonlinear problems is nowadays quite extensive, which includes several important contributions in recent years, most of the main ideas and associated techniques employed can be found in [3], [4], [14], [17], [30], [52], [54], and the references therein. In particular, the first corresponding results for elliptic partial differential equations of second order, which consider a-posteriori error estimators of explicit residual type, the solution of local problems, and the eventual derivation of reliability and efficiency properties, among other issues, go back to [52], [4], [14] and [17].…”

Section: Introductionmentioning

confidence: 99%

A mixed finite element method for Darcy’s equations with pressure dependent porosity

2015

Self Cite

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In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy's equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

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“…The superconvergence-based (averaging-type) error estimators are proposed in [5,8] to control the L 2 -error of the flux variable. Further, the estimators based on the solution of local problems are given in [2,11,14] and a hierarchical estimator can be found in [22]. Finally, a comparison of these four types of error estimators for mixed finite element discretizations by Raviart-Thomas elements is presented in [22].…”

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confidence: 99%

Functional-type <I>a posteriori</I> error estimates for mixed finite element methods

Repin

Smolianski

2005

rus j numer anal math mod

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This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet-Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart-Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging. Functional-type a posteriori error estimates for mixed finite element methods S. I. REPIN * and A. SMOLIANSKI †Abstract -This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet-Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart-Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging. Prof. Yu. A. Kuznetsov This paper is dedicated to the anniversary of

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A posteriori error estimates for the mixed finite element method with Lagrange multipliers

Cited by 18 publications

References 39 publications

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

A mixed finite element method for Darcy’s equations with pressure dependent porosity

Functional-type <I>a posteriori</I> error estimates for mixed finite element methods

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