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 - <p style="text-align: justify;">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.</p>