Journal ArticleDOI
A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations
G. Goodsell,John Robert Whiteman +1 more
TLDR
In this article, a method for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h2) was proposed, both at element edge midpoints and element vertices, using simple averaging schemes over adjacent elements.Abstract:
Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H3 (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h2), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h2) estimates for ∇u in the L2-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.read more
Citations
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Journal ArticleDOI
Computer‐based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations
Journal ArticleDOI
Superconvergence and a posteriori error estimation in triangular mixed finite elements
TL;DR: In this paper, the authors proved superconvergence results for the vector variable when lowest order triangular mixed finite elements of the Raviart-Thomas type on uniform triangulations are used.
Journal ArticleDOI
Superconvergence results on mildly structured triangulations
TL;DR: A unified approach is presented in terms of a generic gradient recovery operator which possesses specific properties on rectangular domains and, using the extended Oganesyan–Rukhovets theorem, superconvergence is proved.
Journal ArticleDOI
Superconvergence Postprocessing for Eigenvalues
TL;DR: It is proved that the proposed algorithm has the superconvergence property of the eigenvalues and this improvement is attained at a small computational cost.
Journal ArticleDOI
Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part II: L∞‐error estimates
G. Goodsell,John Robert Whiteman +1 more
TL;DR: In this paper, superconvergence properties of piecewise quadratic finite element approximations on triangular partitions for two-dimensional elliptic problems and systems, including the case of linear elasticity, were derived.
References
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Journal ArticleDOI
Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation
James H. Bramble,Stephen Hilbert +1 more
Journal ArticleDOI
On superconvergence techniques
Michal Křížek,Pekka Neittaanmäki +1 more
TL;DR: In this article, a brief survey of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented, with a particular emphasis on super-convergent schemes for elliptic problems in the plane.
Journal ArticleDOI
Superconvergence phenomenon in the finite element method arising from averaging gradients
Michal Křížek,Pekka Neittaanmäki +1 more
TL;DR: In this article, the authors studied the superconvergence phenomenon when solving a 2nd order elliptic problem by the usual linear elements and showed that the convergence rate of the averaged gradient to an exact gradient in the L 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.
Journal ArticleDOI
Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations
TL;DR: In this article, a simple schema simple for determining the gradients a partir de l'approximation a elements finis triangulaires lineaire par morceaux de la solution d'un probleme elliptique du second ordre.
Journal ArticleDOI
Superconvergent recovery of gradients on subdomains from piecewise linear finite-element approximations
TL;DR: In this article, cut-off functions are used to prove similar superconvergence results over interior subdomains, which allows superconcverage estimates to be derived for problems with solutions of low global regularity, particularly those involving singularities.
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