An adaptive Levin method for complicated domains

Abstract: 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.