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.

我们对基于双曲和三角广义B样条的等几何离散化产生的矩阵进行频谱分析。考虑具有可变系数的二阶微分问题，并通过嵌套和非嵌套广义样条空间的序列进行离散化。我们证明，当矩阵大小趋于无穷大时，渐近谱分布始终存在，并由多项式紧凑地描述为所谓的符号，就像多项式B样条的情况一样。我们在双曲线，三角函数和多项式情况下观察到符号表达之间的强烈相似之处，这导致相应矩阵的光谱特征相似。通过数字示例说明了理论符号分析，