In this paper, we study a new approach in a posteriori error estimation, in
which the numerical error of finite element approximations is estimated in
terms of quantities of interest rather than the classical energy norm. The
se so-called quantities of interest are characterized by linear functionals
on the space of functions to where the solution belongs. We present here t
he theory with respect to a class of elliptic boundary-value problems, and
in particular, show how to obtain accurate estimates as well as upper and l
ower bounds on the error. We also study the new concept of goal-oriented ad
aptivity, which embodies mesh adaptation procedures designed to control err
or in specific quantities, Numerical experiments confirm that such procedur
es greatly accelerate the attainment of local features of the solution to p
reset accuracies as compared to traditional adaptive schemes based on energ
y norm error estimates. (C) 2001 Elsevier Science Ltd. All rights reserved.