Eigenvalue Isogeometric Approximations Based on B-Splines: Tools and Results

In this note, we focus on the spectral analysis of large matrices coming from isogeometric approximations based on B-splines of the eigenvalue problem $$ -(a(x)u'(x))'=\lambda b(x) u(x),\quad \quad x\in (0,1), $$ where u(0) and u(1) are given. When considering the collocation case, global distribution results for the eigenvalues are available in the literature, despite the nonsymmetry of the related matrices. Here we complement such results by providing precise estimates for the extremal eigenvalues and hence for the spectral conditioning of the resulting matrices. In the Galerkin setting, the matrices are symmetric and positive definite and a more complete analysis has been conducted in the literature. In the latter case we furnish a further procedure that gives a highly accurate estimate of all the eigenvalues, starting from the knowledge of the spectral distribution symbol. The techniques involve dyadic decomposition arguments, tools from the theory of generalized locally Toeplitz sequences, and basic extrapolation methods.