Resumen
This paper establishes some superconvergence estimates for finite element solutions of second-order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution. The projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares. In particular, some local superconvergence estimates in the L2 and $L^\infty$ norms are derived for the local projections of the Galerkin finite element solution. The results have two prominent features. First, they are established for any quasi-uniform meshes, which are of practical importance in scientific computation. Second, they are derived on the basis of local properties of the domain and the solution for the second-order elliptic problem. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in practical computing.

Resumen
This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general framework of convergence is established for some multiplicative Schwarz algorithm. The abstract theory is particularly applied to some obstacle problems, which yields a linear convergence for the corresponding Schwarz overlapping domain decomposition method of one and two levels. Numerical experiments are presented to confirm the convergence estimate derived in this paper.