A Preconditioned Conjugate Gradient Method for Nonselfadjoint or Indefinite Orthogonal Spline Collocation Problems

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 lnN | ln |).