This article surveys a general approach to error control and adaptive mesh
design in Galerkin finite element methods that is based on duality principles
as used in optimal control. Most of the existing work on a posteriori error
analysis deals with error estimation in global norms like the ‘energy norm’
or the L2 norm, involving usually unknown ‘stability constants’. However, in
most applications, the error in a global norm does not provide useful bounds
for the errors in the quantities of real physical interest. Further, their sensitivity
to local error sources is not properly represented by global stability constants.
These deficiencies are overcome by employing duality techniques, as
is common in a priori error analysis of finite element methods, and replacing
the global stability constants by computationally obtained local sensitivity
factors. Combining this with Galerkin orthogonality, a posteriori estimates
can be derived directly for the error in the target quantity. In these estimates
local residuals of the computed solution are multiplied by weights which
measure the dependence of the error on the local residuals. Those, in turn,
can be controlled by locally refining or coarsening the computational mesh.
The weights are obtained by approximately solving a linear adjoint problem.
The resulting a posteriori error estimates provide the basis of a feedback process
for successively constructing economical meshes and corresponding error
bounds tailored to the particular goal of the computation. This approach,
called the ‘dual-weighted-residual method’, is introduced initially within an
abstract functional analytic setting, and is then developed in detail for several
model situations featuring the characteristic properties of elliptic, parabolic
and hyperbolic problems. After having discussed the basic properties
of duality-based adaptivity, we demonstrate the potential of this approach by
presenting a selection of results obtained for practical test cases. These include
problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative
transfer, and optimal control. Throughout the paper, open theoretical
and practical problems are stated together with references to the relevant literature.