Abstract.For the known spectral methods (Galerkin, Tau, Collocation) the condition number behaves like 0(N4) (N: maximal degree of polynomials).We introduce a spectral method with an 0(N2) condition number. The advantages with respect to propagation of rounding errors and preconditioning are demonstrated.A direct solver for constant coefficient problems is given. Extensions to variable coefficient problems and first-order problems are discussed. Numerical results are presented, showing the effectiveness of our methods.

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Abstract.For the known spectral methods (Galerkin, Tau, Collocation) the condition number behaves like 0(N4) (N: maximal degree of polynomials).We introduce a spectral method with an 0(N2) condition number. The advantages with respect to propagation of rounding errors and preconditioning are demonstrated.A direct solver for constant coefficient problems is given. Extensions to variable coefficient problems and first-order problems are discussed. Numerical results are presented, showing the effectiveness of our methods.