Interior boundary conditions in the Schwarz alternating method for the Trefftz method

New types of the Schwarz alternating methods (SAMs) with overlapping and non-overlapping are proposed, by using different interior boundary conditions, such as the Dirichlet, the Neumann and the Robin conditions. Those SAMs using different interior boundary conditions are called the mixed SAMs. This paper consists of two parts. In the first part, for a simple continuous model: the Laplacian solutions on a sectorial domain, the convergence rates of the mixed SAMs are derived in detail for different interior boundary conditions. Compared with the classic overlapping SAM with the Dirichlet conditions, some of mixed SAMs with the interior Neumann or Robin conditions will converge faster. Moreover, a number of new and better SAMs than those in the existing literature are explored in this paper. In the second part, for solving Motz's problem, the Trefftz method using particular solutions and the finite difference method (FDM) are combined, and the mixed SAMs are applied to implement the algorithms into parallel. Numerical results by the mixed SAMs are given to support the analysis made from the simple models. Hence, we may employ some mixed SAMs proposed in this paper to speed the SAM convergence rates.