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Examples of meshes of the computational model with different number of holes, hole sizes and locations are shown in Fig. 4. The bistatic RCS has an exact solution for the case of a plane wave impinging on a 2-D cylinder without holes. We consider the case with ka = π where a is the radius and k = 2π /λ is the wave number associated with wavelength of the incoming wave. We employ a perfectly matched layer (PML) to absorb the reﬂected wave in our computation . The comparison of exact solution and numerical solution is shown in Fig.

1) as a coupled system of a semilinear two-point boundary value problem and a nonlinear functional equation and then to discretize the coupled system by Numerov’s method. To solve the resulting discrete problem we develop a linear monotone iterative algorithm by the method of upper and lower solutions and its associated monotone iterations. 1) by using Numerov’s method without ﬁnding the inverse T −1 . 1) is realized. 1) as a coupled system of a semilinear two-point boundary value problem and a nonlinear functional equation and then apply Numerov’s method to discretize the coupled system.

1181 0 6. Concluding remarks In the paper we have discussed the development of adaptive sparse grid methods in the context of stochastic collocation methods for solving partial differential equations with uncertainty. The emphasis has been on identifying methods which delivers maximum accuracy at minimal cost. We ﬁnd that the combination of Gauss–Patterson quadratures and Smolyak sparse grid constructions is an effective way to reach this and results in a computationally robust approach. In particular we conﬁrm, in agreement with related work, that the widely used Clenshaw–Curtis Smolyak based approach may have problems with convergence for certain high-dimensional test functions.