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

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.