Lower bounds of the discretization error for piecewise polynomials

Lin, Qun; Xie, Hehu; Xu, Jinchao

doi:10.1090/S0025-5718-2013-02724-X

Lower bounds of the discretization error for piecewise polynomials
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Math. Comp. 83 (2014), 1-13 Request permission

Abstract:

Assume that $V_h$ is a space of piecewise polynomials of a degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. There exists the well-known upper bound of the approximation error by $V_h$ for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to $V_h$, the upper-bound error estimate is also sharp.

This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.

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