## ABSTRACT

The subject of this chapter is error analysis and adaptivity with the BNM, as applied to 3-D linear elasticity. Please see the introduction to Chapter 10.

Let the BNM (equation (9.24)) for elasticity be written in operator form as:

LBNM (uk, τk) = 0 ; k = 1, 2, 3 (11.1) with the numerical solution (u∗k, τ

∗ k ). Also, the HBNM (equation (9.27)) is

written in operator form as:

LHBNM (uk, τk) = 0 ; k = 1, 2, 3 (11.2) This time, the stress residual is deﬁned from the stress HBNM (equation

(9.27)) as,

rij ≡ residual(σij) = LHBNM (u∗k, τ∗k ) ; k = 1, 2, 3 (11.3) This idea is illustrated in Figure 10.1(a). It has been proved in [96] and [127] for the BIE that, under certain favorable

conditions, real positive constants c1 and c2 exist such that:

c1r ≤ ≤ c2r (11.4) where r is some scalar measure of a hypersingular residual and is a scalar measure of the exact local error. Thus, a hypersingular residual is expected to provide a good estimate of the local error on a boundary element. It should be

mentioned here that the deﬁnitions of the residuals used in [96] and [127] are analogous to, but diﬀerent in detail from, the ones proposed in this chapter.