Isogeometric discretizations with generalized B-splines: symbol-based spectral analysis

We perform a spectral analysis of matrices arising from isogeometric discretizations based on hyperbolic and trigonometric generalized B-splines. Second-order differential problems with variable coefficients are considered and discretized by means of sequences of both nested and non-nested generalized spline spaces. We prove that an asymptotic spectral distribution always exists when the matrix-size tends to infinity and is compactly described by a so-called symbol, just as in the polynomial B-spline case. We observe a strong resemblance between the symbol expressions in the hyperbolic, trigonometric and polynomial cases, which results in similar spectral features of the corresponding matrices. The theoretical symbol analysis is illustrated with numerical examples, and we show how the symbol can be used to make an analytical prediction of spectral discretization errors.