The least-squares Legendre and Chebyshev pseudospectral methods are presented for a first-order system equivalent to a second-order elliptic partial differential equation. Continuous and discrete homogeneous least-squares functionals using Legendre and Chebyshev weights are shown to be equivalent to the H 1 (Ω) norm and Chebyshev-weighted Div-Curl norm over appropriate polynomial spaces, respectively. The spectral error estimates are derived. The block diagonal finite element preconditioner is developed for the both cases. Several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner.

A closed form expression is given for the eigenvalues and eigenvectors of a symmetric tridiagonal matrix of odd order whose diagonal elements are all equal and whose superdiagonal elements alternate between the values c and d. An implicit formula is given for the even order case.

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First-order system least-squares spectral collocation methods are presented for the Stokes equations by adopting the first-order system and modifying the least-squares functionals in [2]. Then homogeneous Legendre and Chebyshev (continuous and discrete) functionals are shown to be elliptic and continuous with respect to appropriate product weighted norms. The spectral convergence is analyzed for the proposed methods with some numerical experiments.