Some new domain decomposition methods based on natural boundary reduction are suggested for overlapping and non-overlapping domains. A two-dimensional scalar wave equation is taken as a model to illustrate these methods. The governing equation is discretized in time, leading to time-stepping scheme, where an exterior elliptic problem has to be solved in each time step. The Dirichlet-Neumann method and Schwartz alternating method are proposed respectively. For the Schwartz alternating method, the convergence of the algorithm and the contraction factor for exterior circular domain are given. Finally, some numerical examples are devoted to illustrate these methods.

A natural boundary reduction initiated and developed by Feng Kang and Yu Dehao^[1~4] is applied to solve the exterior initial boundary value problem of 2D parabolic equation, and a new coupled method of the natural boundary element method(NBEM) and finite element method(FEM) is suggested. The governing equation is first discretized in time, leading to a time-stepping scheme. Second, the natural integral equation over circular domain is given, and the coupling of the NBEM and FEM for the parabolic equation with unbounded domain is studied. Finally, a numerical example is devoted to illustrate this new method.