The Schwarz domain decomposition (DD) method is widely used in the parallel solution of partial differential equations. In 2001, Garbey and Tromeur-Dervout introduced the idea of Aitken acceleration on the classical additive Schwarz DD method . The authors extend the idea of the Aitken-Schwarz DD method to nonmatching grids in heterogeneous DD methods. Here, it is applied to the “parallel solution of the convection-diffusion equation in a domain composed of a subdomain with homogeneous coefficients and a subdomain with heterogeneous coefficients.” The subdomains are not overlapping. The discretization methods are mixed finite element (MFE) methods (for the heterogeneous coefficient subdomain with an unstructured triangular grid) and spectral element methods (for the homogeneous coefficient subdomain with a rectangular mesh).
The spectral element system is solved by the generalized conjugate residual iterative method with an appropriate preconditioning. The MFE system is “solved by the preconditioned [generalized minimal residual (GMRES)] method with the modified Gram-Schmidt orthogonalization.” A technique to match the two discretization schemes is introduced. Several examples are given to show the robustness of the method. The results show that “computers with a distributed memory and a large number of processors equipped with the message passing interface are affordable for the parallel solutions of large MFE systems.” On the other hand, the spectral element system involves global data, so one is limited to a small number of processors.