S.P.: On the computation of highly oscillatory multivariate integrals with stationary points

Abstract. We consider two types of highly oscillatory bivariate integrals with a nondegenerate critical point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the critical point and by the geometry of the boundary of the underlying domain.