Abstract
Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations.
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Communicated by L. Eldén.
Supported by Communications and Information Technology Ontario (CITO), Canada.
Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
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Christara, C.C., Smith, B. Multigrid and multilevel methods for quadratic spline collocation. Bit Numer Math 37, 781–803 (1997). https://doi.org/10.1007/BF02510352
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DOI: https://doi.org/10.1007/BF02510352