 
Summary: SIAM J. NUMER. ANAL. c 2005 Society for Industrial and Applied Mathematics
Vol. 43, No. 2, pp. 686706
MULTILEVEL PRECONDITIONERS FOR NONSELFADJOINT OR
INDEFINITE ORTHOGONAL SPLINE COLLOCATION PROBLEMS
RAKHIM AITBAYEV
Abstract. Efficient numerical algorithms are developed and analyzed that implement symmetric
multilevel preconditioners for the solution of an orthogonal spline collocation (OSC) discretization of a
Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator. The OSC solution
is sought in the Hermite space of piecewise bicubic polynomials. It is proved that the proposed
additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of
the normal OSC equation. The preconditioners are used with the preconditioned conjugate gradient
method, and numerical results are presented that demonstrate their efficiency.
Key words. orthogonal spline collocation, multilevel methods, preconditioner, nonselfadjoint
or indefinite operator, elliptic boundary value problem
AMS subject classifications. 65N35, 65N55, 65F10
DOI. 10.1137/040609884
1. Introduction. Let be a unit square (0, 1) × (0, 1) with boundary , and
let x = (x1, x2). We consider a Dirichlet boundary value problem (BVP)
Lu = f in and u = 0 on ,(1.1)
where we let
