Superconvergence results on mildly structured triangulations

Lakhany, Asif; Marek, Ivo; Whiteman, J. R.

doi:10.1016/s0045-7825(99)00281-9

Cited by 35 publications

(22 citation statements)

References 26 publications

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“…In recent years there have been some superconvergence results for a recovered gradient [5,15,8,16,20,23,24,25,26,28]. All of them assumed at least u ∈ H 3 (Ω) ∩ W 2 ∞ (Ω) (a condition that rules out domains with a re-entrant corner) and required some stronger (than we required here) mesh conditions.…”

Section: Introduction Let ω ⊂ Rmentioning

confidence: 99%

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?

Wu¹,

Zhang²

2007

SIAM J. Numer. Anal.

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Abstract. We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two-dimensional Poisson equation. Results of this paper are twofold. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered gradient by the polynomial preserving recovery (PPR) is superconvergent. Second, we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on PPR does produce adaptive meshes that asatisfy our mesh density assumption, and the recovered gradient by PPR is indeed superconvergent in the adaptive process.

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Section: Introduction Let ω ⊂ Rmentioning

confidence: 99%

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?

Wu¹,

Zhang²

2007

SIAM J. Numer. Anal.

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“…The study of superconvergence and a posteriori error estimates has been an area of active research; see the monographs by Verfürth [17], Chen and Huang [8], Wahlbin [18], Lin and Yan [16], and Babuška and Strouboulis [3] and a recent article by Lakhany, Marek, and Whiteman [13] for overviews of the field. In this two-part work we study some new superconvergence results.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence

Bank¹,

Xu²

2003

SIAM J. Numer. Anal.

182

194

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In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution u h and to the gradient of the interpolant u I . We then analyze a postprocessing gradient recovery scheme, showing that Q h ∇u h is a superconvergent approximation to ∇u. Here Q h is the global L 2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators. 1. Introduction. The study of superconvergence and a posteriori error estimates has been an area of active research; see the monographs by Verfürth [17], Chen and Huang [8], Wahlbin [18], Lin and Yan [16], and Babuška and Strouboulis [3] and a recent article by Lakhany, Marek, and Whiteman [13] for overviews of the field. In this two-part work we study some new superconvergence results. In Part I, we develop some superconvergence results for finite element approximations of a general class of elliptic partial differential equations (PDEs), based mainly on the geometry of the underlying triangular mesh. In Part II, we develop a gradient recovery technique that can force superconvergence on general shape regular meshes. Patch recovery techniques have been studied by Zienkiewicz and Zhu and this subject has itself evolved into an active subfield of research [25,14,23,24,9,22]. Although our algorithm in some respects resembles this and other similar schemes [12,19,4,6,2,10], it draws much of its motivation from multilevel iterative methods.Let Ω ⊂ R 2 be a bounded domain with Lipschitz boundary ∂Ω. For simplicity of exposition, we assume that Ω is a polygon. We assume that Ω is partitioned by a shape regular triangulation T h of mesh size h ∈ (0, 1). Let V h ⊂ H 1 (Ω) be the corresponding continuous piecewise linear finite element space associated with this triangulation T h , and u h ∈ V h be a finite element approximation to a second order elliptic boundary value problem.Our development has three main steps. In the first step, we prove a superconvergence result for |u h − u I | 1,Ω , where u I is the piecewise linear interpolant for u. In

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“…parallelogram, which is the same as in previous works (see Chen & Huang, 1995;Lakhany et al, 2000;Li & Zhang, 1999;Huang & Xu, 2008;Naga & Zhang, 2004Naga et al, 2006;Wahlbin, 1995;Xu & Zhang, 2003;Zhu & Lin, 1989). When e is in a neighbourhood of O, if h e r 1+μ(1−α)/α e then the condition (α, σ, μ) is fulfilled by those edges such that Ω e is an O(h e ) parallelogram, which means that there is no restriction at all.…”

Section: Mesh Qualitymentioning

confidence: 87%