UN CO RR EC TE D PR O O F 1 Overlapping Domain Decomposition : Convergence 2 Proofs 3

During the last two decades many domain decomposition algorithms have been con8 structed and lot of techniques have been developed to prove the convergence of the 9 algorithms at the continuous level. Among the techniques used to prove the conver10 gence of classical Schwarz algorithms, the first technique is the maximum principle 11 used by Schwarz. Adopting this technique M. Gander and H. Zhao proved a conver12 gence result for n-dimensional linear heat equation in [4]. The second technique is 13 that of the orthogonal projections, used by P. L. Lions in [7], and his convergence 14 results are for linear Laplace equation and linear Stokes equation. In the same pa15 per, P. L. Lions also proved that the Schwarz sequences for linear elliptic equations 16 are related to classical minimization methods over product spaces and this technique 17 was then used by L. Badea in [1] for nonlinear monotone elliptic problems. Another 18 technique is the Fourier and Laplace transforms used in the papers [3, 5] for some 19 1-dimensional evolution equations, with constant coefficients. In [10, 11], S. H. Lui 20 used the idea of upper-lower solutions methods to study the convergence problem for 21 some PDEs, with initial guess to be an upper or lower solution of the equations and 22 monotone iterations. For nonoverlapping optimized Schwarz methods, P. L. Lions 23 in [8] proposed to use an energy estimate argument to study the convergence of the 24 algorithm. The energy estimate technique was then developed in [2] for Helmholtz 25 equation and it has then become a very powerful tool to study nonoverlapping prob26 lems. J.-H. Kimn in [6] proved the convergence of an overlapping optimized Schwarz 27 method for Poisson’s equation with Robin boundary data and S. Loisel and D. B. 28 Szyld in [9] extended the technique of J.-H. Kimn to linear symmetric elliptic equa29 tion. Another technique is to use semiclassical analysis, which works for overlapping 30 optimized Schwarz methods with rectangle subdomains, linear advection diffusion 31 equations on the half plane (see [12]). This paper is devoted to the study of the con32 vergence of Schwarz methods at the continuous level. We give a sketch of the proof 33 of the convergence of optimized Schwarz methods for semilinear parabolic equa34 tions, with multiple subdomains. Complete convergence proofs for both classical 35