Isogeometric spectral approximation for elliptic differential operators

TL;DR

This paper introduces optimized blended quadrature rules for isogeometric spectral approximation of elliptic eigenvalue problems, significantly enhancing accuracy and convergence in structural vibration and quantum mechanics simulations.

Contribution

It generalizes recent methods by developing optimal blending rules that reduce dispersion error and achieve super-convergence in eigenvalue approximations.

Findings

Blended rules improve spectral approximation accuracy.

Optimal blending minimizes dispersion error.

Numerical examples confirm enhanced performance.

Abstract

We study the spectral approximation of a second-order elliptic differential eigenvalue problem that arises from structural vibration problems using isogeometric analysis. In this paper, we generalize recent work in this direction. We present optimally blended quadrature rules for the isogeometric spectral approximation of a diffusion-reaction operator with both Dirichlet and Neumann boundary conditions. The blended rules improve the accuracy and the robustness of the isogeometric approximation. In particular, the optimal blending rules minimize the dispersion error and lead to two extra orders of super-convergence in the eigenvalue error. Various numerical examples (including the Schr$\ddot{\text{o}}$dinger operator for quantum mechanics) in one and three spatial dimensions demonstrate the performance of the blended rules.