Efforts to develop sinc domain decomposition methods for second-order two-point boundary-value problems have been successful, thus warranting further development of these methods. A logical first step is to thoroughly investigate the extension of these methods to Poisson's equation posed on a rectangle. The Sinc-Galerkin and sinc-collocation methods are, for appropriate weight choices, identical for Poisson's equation, and thus only the Sinc-Galerkin system is discussed here. Both the Sincaalerkin patching method and the Sincaalerkin overlapping method are presented in the simple case of decomposition into two subdomains. Numerical results are presented for each of these methods, showing the convergence. As an indication of the capabilities of these domain d e c o m psition techniques, they are applied to Poisson's equation on an L-shaped domain. Restrictions due to the method by which the discrete system is developed require that this problem be solved using nonoverlapping subdomains. Thus only the Sinc-Galerkin patching method is presented. Numerical results are presented that show the convergence of the approximate solutions, even in the presence of boundary singularities.