  In this article we develop convergence theory for a class of goal-oriented
adaptive finite element algorithms for second order nonsymmetric linear
elliptic equations. In particular, we establish contraction results for a
method of this type for Dirichlet problems involving the elliptic operator L u
= div (A grad u) - (b,grad u) - cu, with A Lipschitz, almost-everywhere
symmetric positive definite, with b divergence-free, and with c >= 0. We first
describe the problem class and review some standard facts concerning conforming
finite element discretization and error-estimate-driven adaptive finite element
methods (AFEM). We then describe a goal-oriented variation of standard AFEM
(GOAFEM). Following the recent work of Mommer and Stevenson for symmetric
problems, we establish contraction of GOAFEM and convergence in the sense of
the goal function. Our analysis approach is signficantly different from that of
Mommer and Stevenson, combining the recent contraction frameworks developed by
Cascon, Kreuzer, Nochetto and Siebert; by Nochetto, Siebert and Veeser; and by
Holst, Tsogtgerel and Zhu. We include numerical results demonstrating
performance of our method with standard goal-oriented strategies on a
convection problem.
