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What is the smallest possible constant in Céa’s lemma?

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Abstract

We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝd with d ∈ {1, 2, 3, ...} The constant C ⩾ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to \(1 + \mathcal{O}(h)\) as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.

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Authors and Affiliations

  1. Department of Mathematics, School of Economics, Shandong University, 27 Shanda Nanlu, Jinan, 250 100, P. R. China

    Wei Chen

  2. Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67, Prague 1, Czech Republic

    Michal Křížek

Authors
  1. Wei Chen
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  2. Michal Křížek
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Additional information

This research was supported by Shandong Province Young Scientists Foundation of China 2005BS01008, Institutional Research Plan AV02 101 90503, and by Grant No A 1019201 of the Academy of Sciences of the Czech Republic.

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Chen, W., Křížek, M. What is the smallest possible constant in Céa’s lemma?. Appl Math 51, 129–144 (2006). https://doi.org/10.1007/s10492-006-0009-7

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  • DOI: https://doi.org/10.1007/s10492-006-0009-7

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