Isogeometric Solution of Helmholtz Equation with Dirichlet Boundary Condition in Regions with Irregular Boundary: Numerical Experiences

. In this paper we use the Isogeometric Analysis (IgA) to solve the Helmholtz equa- tion 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 implementation 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.