Year: 2018
Author: So-Hsiang Chou, Tong Sun
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 3 : pp. 392–404
Abstract
A piecewise smooth numerical approximation should be in some sense as smooth as its target function in order to have the optimal order of approximation measured in Sobolev norms. In the context of discontinuous finite element approximation, that means the shape function needs to be numerically smooth in the interiors as well as across the interfaces of elements. In previous papers [2, 8] we defined the concept of numerical smoothness and stated the principle: numerical smoothness is necessary for optimal order convergence. We proved this principle for discontinuous piecewise polynomials on $\mathbb{R}^n$, $1 ≤ n ≤ 3$. In this paper, we generalize it to include discontinuous piecewise non-polynomial functions, e.g., rational functions, on quadrilateral subdivisions whose pullbacks are polynomials such as bilinears, bicubics and so on.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2018-IJNAM-12522
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 3 : pp. 392–404
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Adaptive algorithm discontinuous Galerkin numerical smoothness optimal order convergence.