Remarks about the matrices relative to the Pseudospectral approximation of Neumann problems

S. Bertoluzza, D. Funaro

A standard way to approximate the solution of differential problems is to use collocation methods based on nodes relative to the Jacobi polynomials, such as Chebyshev or Legendre polynomials. The matrices corresponding to the 1D classical differential operatos are known to be full, non symmetric and ill conditioned. We examine those relative to the discretization of Neumann problems in one space dimension. In particular we are concerned with their approximation properties, their eigenvalues and the possibility to find appropriate preconditioners.
Several choices are available when imposing boundary conditions in the approximate problem. For instance they can be either directly enforced or imposed in a variational way. This results in different behaviours, which can drastically affect the numerical treatment of the matrices. We analyze the different cases, pointing out the advantages and the drawback in each strategy. A particular attention is paid to the study of preconditioning matrices

Proc. of IMACS Int. Symp. on ``Iterative Methods in Linear Algebra'', Brussels, 2-4 April 1991, R. Beauwens and P. de Groen eds., North Holland

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