Spectral Analysis of Geometric Multigrid Methods for Isogeometric Analysis
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.
This work was supported by the National Research Network “Geometry + Simulation” (NFN S117, 2012–2016), funded by the Austrian Science Fund (FWF). The first author was also supported by the project AComIn “Advanced Computing for Innovation”, grant 316087, funded by the FP7 Capacity Programme “Research Potential of Convergence Regions”.
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