Can the Discretization Modify the Performance of Schwarz Methods?

Decomposition Methods in Science and Engineering XV, Lecture Notes in Computational Science and Engineering, Springer-Verlag.Schwarz domain decomposition methods can be analyzed both at the continuous and discrete level. For consistent discretizations, one would naturally expect that the discretized method performs as predicted by the continuous analysis. We show in this short note for two model problems that this is not always the case, and that the discretization can both increase and decrease the convergence speed predicted by the continuous analysis