Eigenvalue isogeometric approximations based on B-splines: tools and results. (English) Zbl 1429.65167

Ekström, Sven-Erik; Serra-Capizzano, Stefano

Giannelli, Carlotta (ed.) et al., Advanced methods for geometric modeling and numerical simulation. Selected contributions from the INdAM workshop “DREAMS”, Rome, Italy, January 22–26, 2018. Cham: Springer. Springer INdAM Ser. 35, 57-76 (2019).

Summary: 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.
For the entire collection see [Zbl 1426.65001].

MSC:

65L15	Numerical solution of eigenvalue problems involving ordinary differential equations
65D07	Numerical computation using splines
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations