Optimized Overlapping Domain Decomposition: Convergence Proofs

During the last two decades many domain decomposition algorithms have been constructed and lot of techniques have been developed to prove the convergence of the algorithms at the continuous level. Among the techniques used to prove the convergence of classical Schwarz algorithms, the first technique is the maximum principle used by Schwarz. Adopting this technique M. Gander and H. Zhao proved a convergence result for n-dimensional linear heat equation in Gander and Zhao [2002]. The second technique is that of the orthogonal projections, used by P. L. Lions in Lions [1988], and his convergence results are for linear Laplace equation and linear Stokes equation. In the same paper, P. L. Lions also proved that the Schwarz sequences for linear elliptic equations are related to classical minimization methods over product spaces and this technique was then used by L. Badea in Badea [1991] for nonlinear monotone elliptic problems. Another technique is the Fourier and Laplace transforms used in the papers Giladi and Keller [2002], Gander and Stuart [1998] for some 1-dimensional evolution equations, with constant coefficients. In Lui [2002], Lui [2001], S. H. Lui used the idea of upper-lower solutions methods to study the convergence problem for some PDEs, with initial guess to be an upper or lower solution of the equations and monotone iterations. For nonoverlapping optimized Schwarz methods, P. L. Lions in Lions [1989] proposed to use an energy estimate argument to study the convergence of the algorithm. The energy estimate technique was then developed in Benamou and Desprès [1997] for Helmholtz equation and it has then become a very powerful tool to study nonoverlapping problems. J.-H. Kimn in Kimn [2005] proved the convergence of an overlapping optimized Schwarz method for Poisson’s equation with Robin boundary data and S. Loisel and D. B.