A posteriori error analysis for two non-overlapping domain decomposition techniques

This paper is devoted to the construction of fast solvers for penalty domain decomposition techniques, based upon a posteriori error analysis. We introduce a penalty non-overlapping domain decomposition method (ddm) motivated by the a posteriori error analysis of the method proposed by Chacón and Chacón in [T. Chacón Rebollo, E. Chacón Vera, A non-overlapping domain decomposition method for the Stokes equations via a penalty term on the interface, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1–16]. In the new method a penalty term replaces the L2(Γ) one in the original method. The number of iterations needed by the new ddm to yield a solution with an error of the same order as the discretization error is remarkably reduced. We develop an a posteriori error analysis that we use to determine an optimal value of the penalty parameter for a given grid, and also to jointly determine an optimal grid and a penalty parameter to reduce the error below a targeted value. Several numerical tests for model problems exhibit the good performances of our approach and provide to a numerical comparison of the two penalty methods.