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

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

Published online: 2018-03

Cited by

Export citation
• 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.

65M12, 65M15, 65N30.

tsun@bgsu.edu (Tong Sun)

• BibTex
• RIS
• TXT
@Article{IJNAM-15-392, author = {So-Hsiang and Chou and and 20699 and and So-Hsiang Chou and Tong and Sun and tsun@bgsu.edu and 7974 and Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403-0221, USA and Tong Sun}, 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.

So-Hsiang Chou & Tong Sun. (2020). Optimal Order Convergence Implies Numerical Smoothness II: The Pullback Polynomial Case. International Journal of Numerical Analysis and Modeling. 15 (3). 392-404. doi:
Copy to clipboard
The citation has been copied to your clipboard