Optimal spectral approximation of $2n$-order differential operators by mixed isogeometric analysis

TL;DR

This paper develops an improved mixed isogeometric analysis method for accurately approximating the spectra of high-order differential operators, showing enhanced convergence and spectral accuracy over traditional finite element approaches.

Contribution

It introduces an optimal quadrature rule application in mixed isogeometric analysis, achieving higher spectral approximation accuracy for complex differential operators.

Findings

Eigenvalue error convergence order improved to 2p+2 with optimal quadrature.

Mixed isogeometric analysis outperforms mixed finite element methods in spectral approximation.

Numerical results confirm the theoretical convergence improvements.

Abstract

We approximate the spectra of a class of $2n$-order differential operators using isogeometric analysis in mixed formulations. This class includes a wide range of differential operators such as those arising in elliptic, biharmonic, Cahn-Hilliard, Swift-Hohenberg, and phase-field crystal equations. The spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations, which require auxiliary parameters. The mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order $2p$ where $p$ is the order of the underlying B-spline space. We improve this order to be $2p+2$ by applying optimally-blended quadrature rules developed in \cite{20,52} and this order is an optimum in the view of dispersion error. We also compare these results with the mixed finite elements and show numerically that mixed isogeometric analysis leads to significantly better spectral approximations.