  This paper is concerned with a numerical solution of the acoustic scattering
by a bounded impenetrable obstacle in three dimensions. The obstacle scattering
problem is formulated as a boundary value problem in a bounded domain by using
a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is
derived for the finite element method with the truncated DtN operator. The a
posteriori error estimate consists of the finite element approximation error
and the truncation error of the DtN operator, where the latter is shown to
decay exponentially with respect to the truncation parameter. Based on the a
posteriori error estimate, an adaptive finite element method is developed for
the obstacle scattering problem. The truncation parameter is determined by the
truncation error of the DtN operator and the mesh elements for local refinement
are marked through the finite element approximation error. Numerical
experiments are presented to demonstrate the effectiveness of the proposed
method.
