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Volume 15, Issue 3
Optimal Order Convergence Implies Numerical Smoothness II: The Pullback Polynomial Case

So-Hsiang Chou & Tong Sun

Int. J. Numer. Anal. Mod., 15 (2018), pp. 392-404.

Published online: 2018-03

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  • 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.

  • Keywords

Adaptive algorithm, discontinuous Galerkin, numerical smoothness, optimal order convergence.

  • AMS Subject Headings

65M12, 65M15, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

tsun@bgsu.edu (Tong Sun)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-392, author = {Chou , So-Hsiang and Sun , Tong}, title = {Optimal Order Convergence Implies Numerical Smoothness II: The Pullback Polynomial Case}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {3}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12522.html} }
TY - JOUR T1 - Optimal Order Convergence Implies Numerical Smoothness II: The Pullback Polynomial Case AU - Chou , So-Hsiang AU - Sun , Tong JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 392 EP - 404 PY - 2018 DA - 2018/03 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12522.html KW - Adaptive algorithm, discontinuous Galerkin, numerical smoothness, optimal order convergence. AB -

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.

Chou , So-Hsiang and Sun , Tong. (2018). Optimal Order Convergence Implies Numerical Smoothness II: The Pullback Polynomial Case. International Journal of Numerical Analysis and Modeling. 15 (3). 392-404. doi:
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