Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

Yan, Ningning

doi:10.1007/s10492-009-0017-5

Cited by 33 publications

(29 citation statements)

References 21 publications

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“…One is based on the analysis of the method: there are the integral identity, the integral expansion (which is based on the Bramble-Hilbert lemma) and others; see [8,35]. The second concerns the mesh condition [2,23,36,37,38,40,44]. The supercloseness result has been extended from structured meshes to more general, practical and automatically generated meshes.…”

Section: Introductionmentioning

confidence: 99%

Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem

2011

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Abstract. The supercloseness and superconvergence properties of stabilized finite element methods applied to the Stokes problem are studied. We consider consistent residual based stabilization methods as well as inconsistent local projection type stabilizations. Moreover, we are able to show the supercloseness of the linear part of the MINI-element solution which has been previously observed in practical computations. The results on supercloseness hold on three-directional triangular, axiparallel rectangular, and brick-type meshes, respectively, but extensions to more general meshes are also discussed. Applying an appropriate postprocess to the computed solution, we establish superconvergence results. Numerical examples illustrate the theoretical predictions.

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Section: Introductionmentioning

confidence: 99%

Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem

2011

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“…[12,28]. But we have a long series of lemmas dealing with each term in the bilinear forms a(·, ·) and b(·, ·).…”

Section: Superclosenessmentioning

confidence: 99%

Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

Huang

Zhang

2013

Commun. Math. Stat.

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By the standard theory, the stable Q k+1,k − Q k,k+1 /Q dc k divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H 1 -norm and the pressure in L 2 -norm. This is due to one polynomial degree less in y direction for the first component of velocity, which is a Q k+1,k polynomial of x and y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k + 1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

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“…The following supercloseness for bilinear elements can be found, e.g., in [6], [10], p. 314, [18], p. 9,…”

Section: Remark 32mentioning

confidence: 99%

Two-sided bounds of the discretization error for finite elements

Křížek¹,

Roos²,

Wei³

2011

ESAIM: M2AN

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Abstract.We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.Mathematics Subject Classification. 65N30.

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Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

Cited by 33 publications

References 21 publications

Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem

Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem

Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

Two-sided bounds of the discretization error for finite elements

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