A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions

In our earlier work [4], a dual iterative substructuring method for two dimensional problems was proposed, which is a variant of the FETI-DP method. The proposed method imposes continuity on the interface by not only the pointwise matching condition but also uses a penalty term which measures the jump across the interface. For a large penalization parameter, it was proven that the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. In this paper, we extend the method to three dimensional problems. For this extension, we consider two things; one is the construction of a penalty term in 3D to give the same convergence speed as in 2D and the other is how to treat the ill-conditioning of the subdomain problems due to a large penalization parameter. To resolve these two key issues, we need to be aware of the difference between 2D and 3D in the geometric complexity of the interface. Based on the geometric observation for the difference, we suggest a modified penalty term and a preconditioner aiming at reducing couplings between functions on the interface.