Comparison of round-off errors in integral equation formulations of elastostatical boundary value problems

Abstract

The two cardinal boundary value problems of elastostatics can be formulated both as integral equations of the first and of the second kind. However, in nearly all cases the problem with prescribed boundary displacements is solved with the aid of an equation of the first kind and the problem with prescribed boundary tractions with the aid of an equation of the second kind. For numerical treatment, the integral equation is usually approximated by a system of algebraic equations. The resultant error is called the discretization error or the interpolation error. Since computers do not perform arithmetic operations with an infinite number of digits, the numerical solution also contains round-off errors. The interpolation error decreases with the fineness of the discretization, i.e. with the number of algebraic equations, whereas the round-off error increases. Hence, as the total error has a minimum it is not reasonable in certain cases to try to increase the accuracy of the results by refining the division of the boundary into elements. Usually the interpolation error is dominant. However, if the coefficient matrices are large or if the calculatory operations are performed with a small number of digits the round-off error may prevail. In this paper we investigate, by examples, the circumstances under which results are decisively affected by round-off error. The numerical conditioning of integral equations of the first kind is worse than that of equations of the second kind. In the first case the condition numbers of the coefficient matrices tend to infinity with increasing numbers of algebraic equations, whereas in the second case they tend to a finite value. Therefore, theoreticians warn against using integral equations of the first kind. Most practitioners are of a different opinion. They prefer integral equations of the first kind to equations of the second kind for the solution of the problem with prescribed displacements. The adequate accuracy of the results obtained by using these equations seems to vitiate the theoretical arguments. This paper tries to weigh up the pros and cons of both points of view. The different behavior of the round-off errors of the solutions of integral equations of the first and of the second kind and the consequences for the treatment of technical problems are investigated. (For an extensive survey of this paper: see end of section 1.)