Resumen
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 H1(\Omega)$ 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.

Resumen
In this work we consider the semicirculant preconditioning of elliptic differential operators of the form Lu := - \epsilon \Delta u + au_x + bu_y + cu $$ in two cases: $0 < \epsilon \ll 1$ and $\epsilon \equiv 1$. The paper [Numer. Math., 81 (1998), pp. 211--249] provided extremely interesting and useful results in the first case. On the other hand, those appear to contradict basic results on preconditioning given in [SIAM J. Numer. Anal., 27 (1990), pp. 656--694]. We reobtain the results of [Numer. Math., 81 (1998), pp. 211--249] by a new approach which we believe to be more transparent. We also clarify the situation regarding the apparent contradiction with [SIAM J. Numer. Anal., 27 (1990), pp. 656--694]. Finally, we describe the distribution of the preconditioned eigenvalues in the uniformly elliptic case, $\epsilon \equiv 1$.