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Remarks about the matrices relative to the pseudospectral approximation of Neumann problems. (English) Zbl 0785.65087
Beauwens, R. (ed.) et al., Iterative methods in linear algebra. Proceedings of the IMACS international symposium, Brussels, Belgium, 2-4 April, 1991. Amsterdam: North-Holland. 209-215 (1992).
Summary: A standard way to approximate the solution of differential problems by algebraic polynomials is to use collocation methods based on nodes related to Jacobi polynomials, such as Chebyshev or Legendre polynomials. The matrices corresponding to the 1D classical differential operators are known to be full, nonsymmetric 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 behaviors, which can drastically affect the numerical treatment of the matrices. We analyze the different cases, pointing out the advantages and the drawbacks in each strategy. A particular attention is paid to the study of preconditioning matrices.
For the entire collection see [Zbl 0778.00018].

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65F35 Numerical computation of matrix norms, conditioning, scaling