A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with τ-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for p = 1, 2, 3. For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.

Two-grid optimality for Galerkin linear systems based on B-splines

DONATELLI, MARCO;SERRA CAPIZZANO, STEFANO;
2015-01-01

Abstract

A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with τ-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for p = 1, 2, 3. For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.
2015
http://www.springerlink.com/content/r6x468507522/
B-splines; Isogeometric analysis; Multigrid methods; τ-matrices; Modeling and Simulation; Theoretical Computer Science; Software; 1707; Computational Theory and Mathematics; Engineering (all)
Donatelli, Marco; Garoni, C.; Manni, C.; SERRA CAPIZZANO, Stefano; Speleers, H.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2047969
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