Spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness

We present a first step towards the spectral analysis of matrices arising from IgA Galerkin methods based on hyperbolic and trigonometric GB-splines. Second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non-nested spline spaces. We prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol, just like in the polynomial case. There is a complete similarity between the symbol expressions in the hyperbolic, trigonometric and polynomial cases. This results in similar spectral features of the corresponding matrices. We also analyze the IgA discretization based on trigonometric GB-splines for the eigenvalue problem related to the univariate Laplace operator. We prove that, for non-nested spaces, the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case. As part of the analysis, we derive several Fourier properties of cardinal GB-splines.