Resumen
A nonoverlapping domain decomposition iterative procedure is developed and analyzed for second order elliptic problems in $\mathbb R^N$. Its convergence is proved. The method is based on a Robin-type consistency condition with two parameters, called a transmission coefficient and a penalty coefficient, as a transmission condition together with a derivative-free transmission data updating technique on the artificial interfaces. Then the method is applied to the nonconforming finite element problems. A nonoverlapping domain decomposition iterative procedure for solving the nonconforming finite element problems of second order partial differential equations is developed and analyzed, which is directly presented to the nonconforming finite element problems without introducing any Lagrange multipliers. Its convergence is demonstrated, and the convergence rate is derived. The convergence analyses imply that the convergence rate is independent of the finite element meshes size while choosing the right parameters. Furthermore, the conclusions are extended to the unstructured finite element meshes. For both continuous problems and discrete problems, the method of this paper can be applied to general multisubdomain decompositions and implemented on parallel machines with local communications naturally. The method also allows choosing subdomains very flexibly, even as small as an individual element for finite element problems.

Resumen
The aim of this paper is to develop stepwise variable preconditioning for the iterative solution of monotone operator equations in Hilbert space and apply it to nonlinear elliptic problems. The paper is built up to reflect the common character of preconditioned simple iterations and quasi-Newton methods. The main feature of the results is that the preconditioners are chosen via spectral equivalence. The latter can be executed in the corresponding Sobolev space in the case of elliptic problems, which helps both the construction and convergence analysis of preconditioners. This is illustrated by an example of a preconditioner using suitable domain decomposition.