PRECONDITIONED RICHARDSON AND MINIMAL RESIDUAL ITERATIVE METHODS FOR PIECEWISE HERMITE BICUBIC ORTHOGONAL SPLINE COLLOCATION EQUATIONS

The preconditioned Richardson and preconditioned minimal residual iterative methods are presented for the solution of linear equations arising when orthogonal spline collocation with piecewise Hermite bicubics is applied to a selfadjoint elliptic Dirichlet boundary value problem on a rectangle. For both methods, the orthogonal spline collocation discretization of Laplace's operator is used as a preconditioner. In the preconditioned Richardson method, an approximation of the optimal iteration parameter is computed from knowledge of spectral equivalence constants. Such a priori information is not required for the preconditioned minimal residual method. In each iteration of both methods, orthogonal spline collocation Poisson's problems are solved by a fast direct algorithm which employs fast Fourier transforms. An application of the preconditioned minimal residual method is also discussed for the solution of linear equations arising from the orthogonal spline collocation discretization of nonselfadjoint elliptic Dirichlet boundary value problems.