Stability Analysis of Some Non-conforming FETI Domain Decomposition Methods for Electromagnetic Problems

摘要

Non-conforming Domain Decomposition Methods (DDM) have been proven to be very efficient solvers in the analysis of large and repetitive structures in electromagnetics (EM). Although much recent work [1],[2] has been devoted on non-conforming DDM for EM problems, issues related to robustness of the algorithms have not been thoroughly studied. As a form of iterative solver, the success of DDM is sensitive to the conditioning of the final system that among others depends on the formulation, discretization, solution regularity (materials and geometry) and frequency. Numerical stability is the concept that describes the behaviour of the condition number when the discretisation of the problem tends to zero, thus it can provide useful information about the robustness of a formulation. Specifically, this study will focus only on a subset of non-conforming DDM namely the finite element tearing and interconnecting (FETI) formulation because are considerably more efficient that other DDM approaches. The purpose of this paper is to study the stability of a proposed FETI formulation as well as some existing FETI and FETI-like ones. Only cement [1] type non-conforming formulations are considered because they are amenable to FETI and lead to convergent formulations at the continuous level because they enforce Robin transmission conditions. From this stability study it will become clear that no existing cement FETI method is both efficient and stable, whereas cement FETI-like methods [2] although stable are not as efficient as the FETI ones.