A Best Approximation Evaluation of a Finite Element Calculation Page: 2 of 13
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1 Introduction
This paper is a report on the mathematics behind an indicator that is cur-
rently being used to measure the computational quality of a particular finite
element calculation. It is offered as an example of an application of linear
algebra. The study is prompted by a physical problem; the solution is based
upon familiar best approximation techniques.
The physical problem consists of an electrostatic system with n conduct-
ing plates. The charge on each conductor can be related to the voltages on
all the conductors through a linear transformation in the form of an n-by-
n matrix. The physics of the situation require that the solution matrix be
symmetric with a special property: every row sum adds to zero. However, if
the solution is calculated numerically by a finite element method, then the
output matrix A is an approximation to the theoretically correct solution Ae.
In particular, A may not be symmetric nor have zero row and column sums.
This leads to the question: is A an acceptable solution to the problem?
One response to this question is to view the totality S of symmetric,
zero row sum matrices as a subspace of the real n-by-n matrices with an
inner product. The distance between A and Ae provides an indication of
the computational quality of the finite element calculation: that is, A is
considered to be an acceptable approximation to Ae only if the distance
between A and Ae is sufficiently small. Generally, however, this distance is
not known. Nevertheless, it is at least as large as the distance from A to the
closest element Ao of S. In particular, an excessive distance between A and
A0 warns of a possible calculation error or an input or coding problem, and A
should be rejected. In other words, the distance between A and Ao provides a
computational quality indicator, a notion that is central in simulation based
engineering.
In this paper we detail the construction of this particular indicator. Specif-
ically, with respect to the Frobenius inner product, we show how to project
any matrix A onto the closest matrix Ao of the subspace S and to determine
the distance between A and Ao. Sandia National Laboratories has imple-
mented this best approximation method into a production code, where the
algorithm has proven to be both effective and inexpensive.2
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ROBINSON, ALLEN C. & ROBINSON, DONALD W. A Best Approximation Evaluation of a Finite Element Calculation, article, September 29, 1999; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc623866/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.