  In this paper we describe an adaptive Levin method for numerically evaluating
integrals of the form $\int_\Omega f(\mathbf x) \exp(i g(\mathbf x)) \,d\Omega$
over general domains that have been meshed by transfinite elements. On each
element, we apply the multivariate Levin method over adaptively refined
sub-elements, until the integral has been computed to the desired accuracy.
Resonance points on the boundaries of the elements are handled by the
application of the univariate adaptive Levin method. When the domain does not
contain stationary points, the cost of the resulting method is essentially
independent of the frequency, even in the presence of resonance points.
