We investigate geometric multigrid methods for solving the large, sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. We observe that the performance of standard V-cycle iteration is highly dependent on the spatial dimension as well as the spline degree of the discretization space. Conjugate gradient iteration preconditioned with one V-cycle mitigates this dependence, but does not eliminate it. We perform both classical local Fourier analysis as well as a numerical spectral analysis of the two-grid method to gain better understanding of the underlying problems and observe that classical smoothers do not perform well in the isogeometric setting.
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