Goal-oriented error estimation and adaptivity for the finite element method

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.