Multilevel Preconditioners for Non-self-adjoint or Indefinite Orthogonal Spline Collocation Problems

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 non–self-adjoint 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.