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 Puzyrev et al. (2017), Caloet al. (0000) 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 the mixed isogeometric analysis leads to significantly better spectral approximations.
Highlights
The finite element method (FEM) is a widely used and highly effective numerical technique for approximate solutions of boundary value problems
For optimal blending parameters of higher order p and blending among other quadrature rules, we refer to [61]. These optimally-blended quadrature rules improve spectrum errors significantly, which we show
We present and study a mixed formulation of isogeometric analysis for a set of
Summary
The finite element method (FEM) is a widely used and highly effective numerical technique for approximate solutions of boundary value problems. Dispersion-minimizing methods based on modified integration rules for reducing dispersion error have been developed previously for classical FEM [39, 40] and isogeometric analysis [1, 2]. For the standard finite and isogeometric elements, these optimal dispersion methods lead to two additional orders of error convergence (superconvergence) in the eigenvalues, while the eigenfunction errors do not degenerate. We utilize the mixed FEM framework for a general 2n-order linear differential eigenvalue problem. We develop the mixed isogeometric framework for these eigenvalue problems and present error analysis for both eigenvalue and eigenfunctions. Optimal blending rules for the mixed isogeometric discretizations of the 2n-order problem are presented up to n = 3.
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