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Generalizations and Accelerations of Lions' Nonoverlapping Domain Decomposition Method for Linear Elliptic PDE

In this paper, a one-parameter generalization of Lions' nonoverlapping domain decomposition method for linear elliptic PDEs is proposed and studied. The generalized methods are shown to be descent-direction methods for minimizing an interface bias functional. Iteration convergence of both the continuous and finite element versions of the proposed methods is established. It is theoretically and numerically demonstrated that for generic choices of the parameter the generalized methods converge faster than Lions' original method. Algorithms are given and numerical results are presented.