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.

翻译：本文描述了一种自适应Levin方法，用于数值计算形式为$\int_\Omega f(\mathbf x) \exp(i g(\mathbf x)) \,d\Omega$的积分，其中积分域为采用超限元网格化的一般区域。在每个单元上，我们在自适应细分的子单元上应用多元Levin方法，直至积分计算达到所需精度。单元边界上的共振点通过应用一元自适应Levin方法进行处理。当积分域不含驻点时，即使存在共振点，该方法的计算成本本质上与频率无关。