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

So–hsiang Chou & Tong Sun

DOI:

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 Rn, 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

chou@bgsu.edu (So–hsiang Chou)

tsun@bgsu.edu (Tong Sun)

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