A Lions non-overlapping domain decomposition method for domains with an arbitrary interface

Lions' non-overlapping domain decomposition method for the solution of elliptic partial differential equations has been analysed extensively by many authors. There have been numerous works on the convergence of the iterative method as well as variations of it. In the present work, we analyse several formulations of Lions' method. For two of these, we show that the spectral radius of the operator in the fixed-point iteration for the interface boundary function is bounded above by 1 – O(h
^1/2
) when the optimal value (O(h
^–1/2
)) of the parameter in the Robin boundary condition along the artificial interface is used. While this result is already known for rectangular domains with a straight interface, our analysis is valid for essentially arbitrary geometry. The method of Guo and Hou, which is Lions' method with relaxation, has a simple proof of convergence using our method of analysis. Similarly, we provide a simple proof of convergence of the scheme of Deng.