Approximation Order from Bivariate C1-Cubics: A Counter Example.
Abstract
It is shown that the space of bivariate C1 piecewise cubic functions on a hexagonal mesh of size h approximates to certain smooth functions only to within O(h3) even though it contains a local partition of every cubic polynomial. One measures the approximation power of a family S of piecewise polynomial approximating functions on some partition in terms of the meshsize h of that partition. Typically, the error of approximation goes to zero like hr as the meshsize goes to zero, with r depending on the smoothness of the function being approximated. There is a maximal r typical for the space S used, and faster convergence rates are possible only for very special functions. Naturally, one would like this optimal rate or approximation order hr to be as fast as possible, i.e., would like the maximal r to be as large as possible. In any case, it is an important practical question to ascertain, for a given approximating space S, what its optimal approximation order is.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 01, 1982
- Accession Number
- ADA118606
Entities
People
- C. De Boor
- K. Hoellig
Organizations
- University of Wisconsin–Madison