 
Summary: A PRECONDITIONED CONJUGATE GRADIENT METHOD FOR
NONSELFADJOINT OR INDEFINITE ORTHOGONAL SPLINE
COLLOCATION PROBLEMS
RAKHIM AITBAYEV AND BERNARD BIALECKI
SIAM J. NUMER. ANAL. c 2003 Society for Industrial and Applied Mathematics
Vol. 41, No. 2, pp. 589604
Abstract. We study the computation of the orthogonal spline collocation solution of a linear
Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form Lu =
aij(x)uxixj + bi(x)uxi + c(x)u. We apply a preconditioned conjugate gradient method to the
normal system of collocation equations with a preconditioner associated with a separable operator,
and prove that the resulting algorithm has a convergence rate independent of the partition step size.
We solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm.
On a uniform N ×N partition, the cost of the algorithm for computing the collocation solution within
a tolerance is O(N2 ln N ln ).
Key words. nonselfadjoint or indefinite elliptic boundary value problem, orthogonal spline
collocation, conjugate gradient method, preconditioner, matrix decomposition algorithm
AMS subject classifications. 65N35, 65N22, 65F10
PII. S0036142901391396
1. Introduction. On = (0, 1) × (0, 1) with boundary , we consider the
Dirichlet boundary value problem (BVP)
