In this paper we discuss the numerical solution of the Helmholtz equation with mixed boundary conditions on a 2D physical domain Ω. The so called radiation problem depends on the constant wavenumber k, that in some medical applications can be of order of thousands. For these values of k the classical Finite Element Method (FEM) faces up several numerical difficulties. To mitigate these limitations we apply the Isogeometric Analysis (IgA) to compute the approximated solution uh. Main steps of IgA are discussed and specific proposals for their fulfillment are addressed, with focus on some aspects not covered in available publications. In particular, we introduce a low distortion quadratic NURBS parametrization of Ω that represents exactly its boundary and contributes to the accuracy of uh. Our approach is non-isoparametric since uh is a bicubic tensor product polynomial B-spline function on Ω. This allows to improve the numerical solution refining the approximation space and keeping the coarser parametrization of the domain. Moreover, we discuss the role of the number of degrees of freedom in the directions perpendicular and longitudinal to wave front and its relationship with the noise and the shift in amplitude and phase of uh. The linear system derived from IgA discretization of the radiation problem is solved using GMRES and we show through experiments that the incomplete factorization of the Complex Shifted Laplacian provides a very good preconditioner. To solve the radiation problem, we have implemented IgA approach using the open source package GeoPDEs. A comparison with FEM is included, to provide evidence that IgA approach is superior since it is able to reduce significantly the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.@en

In this paper we use the Isogeometric Analysis (IgA) to solve the Helmholtz equation with Dirichlet boundary condition over a bounded physical 2D domain. Starting from the variational formulation of the problem, we show how to apply IgA to obtain an approximated solution based on biquadratic B-spline functions. We focus the attention on problems where the physical domain has very irregular boundary. To solve these problems successfully a high quality parametrization of the domain must be constructed. This parametrization is also a biquadratic tensor product B-spline function, with control points computed as the vertices of a quadrilateral mesh with optimal geometric properties. We study experimentally the influence of the wave number and the parametrization of the physical domain in the accuracy of the approximated solution. A comparison with classical Finite Element Method is also included. The power of IgA is shown solving several difficult model problems, which are particular cases of the Helmholtz equation and where the solution has discontinuous gradient in some points, or it is highly oscillatory. For all model problems we explain how to select the knots of B-spline quadratic functions and how to insert knew knots in order to obtain good approximations. The results obtained with our imple-mentation of the method prove that IgA approach is successful, even on regions with irregular boundary, since it is able to offer smooth solutions having at the same time some singular points and high number of oscillations.

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