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In this paper, optimized Schwarz methods with Robin transmission conditions are considered for solving the time-dependent linear algebraic systems resulting from finite element approximations for parabolic problems. We use both analysis and numerical experiments to investigate the convergence behavior. Particularly, we check the influence of the time step size on the convergence rate. We get the estimate of the upper bound of convergence rate 1 - O(min{h1/2H-1/2, h1/2H3/2t-1}), where h is the mesh size, H is the size of subdomains, and t is the time step size. Our numerical results show that the convergence rate of this method is fast. We also search for the optimal parameter λ by experiments and find the rule of its distribution. [ABSTRACT FROM AUTHOR]

In recent years, a nonoverlapping domain decomposition iterative procedure, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed. It is known that the convergence rate of this method is 1 − O( h), where h is mesh size. In this paper, the convergence rate is improved to be 1 − O( h 1/2H−1/2) sometime by choosing suitable parameter, where H is the subdomain size. Counter examples are constructed to show that our convergence estimates are sharp, which means that the convergence rate cannot be better than 1 − O( h 1/2H−1/2) in a certain case no matter how parameter is chosen. [ABSTRACT FROM AUTHOR]

In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [ABSTRACT FROM AUTHOR]