A Counterexample to a Result of Strang and Fix Concerning Controlled Approximation.
Abstract
In finite element analysis is it is important to choose appropriate trial functions. One approach to such a choice has been described by Strang as follows: Choose one or more locally supported functions phi sub l phi sub n rescale the independent variable, and translate the functions just constructed. A principal question in finite element analysis is how well a given function can then be approximated by those trial functions. This is measured by the order of approximation. Sometimes it is required that the coefficients in the process of approximation be bounded. Such an approximation is said to be controlled. A result of Strang and Fix states that if a collection of locally supported elements has controlled approximation of order k, then there is a finite linear combination omega of those elements and their translates such that any polynomial of degree less than k can be reproduced by omega nd its translates. This result has been questioned for some time. Recently, the rapid development of the theory of multivariate spline functions has given a new impetus to the controlled approximation problem. It becomes significant to answer the question whether their result is correct. This paper gives a counterexample to the result of Strang and Fix. The author's conclusion suggests that the controlled approximation order should be refined. Hopefully, the result in this paper will lead to further development of finite element analysis.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1984
- Accession Number
- ADA147360
Entities
People
- R. Q. Jia
Organizations
- University of Wisconsin–Madison