Fast algorithms for high-order spline collocation systems
Well-known fast direct algorithms have been developed for low-order discretizations to solve elliptic partial differential equations. A low-order discretization often gives a simple matrix structure. Development in this area has been described by \textit{R. W. Hockney} [J. Assoc. Comput. Machin. 12, 95-113 (19965; Zbl 0139.10902)], \textit{F. W. Dorr} [SIAM Rev. 12, 248-263 (1970; Zbl 0208.42403)], \textit{J. R. Rice} and \textit{R. F. Boisert} [Solving elliptic problems using ELLPACK (1985; Zbl 0562.65064)] and others. In this paper the author presents a complete eigenvalue analysis for arbitrary \(C^1\)-spline collocation methods applied to the Poisson equation on a rectangular domain with Dirichlet boundary conditions. Based on this analysis, the author develops some fast algorithms to solve a class of high-order spline collocation systems which arise from discretizing the Poisson equation.
scientific article; zbMATH DE number 1251318