MULTILEVEL ADDITIVE AND MULTIPLICATIVE METHODS FOR ORTHOGONAL SPLINE COLLOCATION PROBLEMS

Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation w ith piecewise Hermite bicubics is applied to the Dirichlet boundary va lue problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are define d respectively as sums and products of independent operators on a sequ ence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the col location discretization of the Laplacian with the spectral constants i ndependent of the fine partition stepsize and the number of levels. Th e preconditioned conjugate gradient and preconditioned Orthomin method s are considered for the solution of collocation problems. An implemen tation of the methods is discussed and the results of numerical experi ments are presented.