Robust and Scalable Domain Decomposition Methods for Electromagnetic Computations

The Finite Element Tearing and Interconnecting (FETI) and its variants are probably the most celebrated domain decomposition algorithms for partial differential equation (PDE) scientific computations. In electromagnetics, such methods have advanced research frontiers by enabling the full-wave analysis and design of finite phased array antennas, metamaterials, and other multiscale structures. Recently, closer scrutiny of these methods have revealed robustness and numerical scalability problems that prevent the most memory and time efficient variants of FETI from gaining widespread acceptance. This work introduces a new class of FETI methods and preconditioners that lead to exponential iterative convergence for a wide class of problems, are robust and numerically scalable. First, a two Lagrange multiplier (LM) variant of FETI with impedance transmission conditions, the FETI-2λ, is introduced to facilitate the symmetric treatment of non-conforming grids while avoiding matrix singularites that occur at the interior resonance frequencies of the domains. A thorough investigation on the approximability and stability of the Lagrange multiplier discrete space is carried over to identify the correct LM space basis. The resulting method, although accurate and flexible, exhibits unreliable iterative convergence. To accelerate the iterative convergence, the Locally Exact Algebraic Preconditioner (LEAP), which is responsible for improving the information transfer between neighboring domains is introduced. The LEAP was conceived by carefully studying the properties of the Dirichlet-to-Neumann (DtN) map that is involved in the sub-structuring process of FETI. LEAP proceeds in a hierarchical way and directly factorizes the signular and near-singular interactions of the DtN map that arise from domain-face, domain-edge and domain-vertex interactions. For problems with small number of domains LEAP results in scalable implementations with respect to the discretization. On problems with large domain numbers, the numerical scalability can only be obtained through ``global\u27\u27 preconditioners that directly convey information to remotely separated domains at every DDM iteration. The proposed ``global preconditiong stage is based on the new Multigrid FETI (MG-FETI) method. This method provides a coarse grid correction mechanism defined in the dual space. Macro-basis functions, that satisfy thecurl-curl equation on each interface are constructed to reduce the size of the coarse problem, while maintaining a good approximation of the characteristic field modes. Numerical results showcase the performance of the proposed method on one-way, 2D and 3D decomposed problems, with structured and unstructured partitioning, conforming and non-conforming interface triangulations. Finally, challenging, real life computational examples showcase the true potential of the method