Nonconformal FETI-DP methods for numerical analysis of large-scale electromagnetic problems

Abstract

The finite element tearing and interconnecting dual primal (FETI-DP) method, as a nonoverlapping domain decomposition method (DDM), was originally developed to model acoustic scattering of submarine objects. Later, it was extended successfully to solving the vector Maxwell's equations for large-scale electromagnetic problems, such as finite antenna arrays, photonic crystal cavities, and array-structured metamaterials, by employing hierarchical vector basis functions. This method employs Lagrange multipliers, which are defined at the subdomain interfaces, to couple the fields in different subdomains. Furthermore, unknowns associated with geometrical crosspoints (shared by three or more subdomains) are extracted to form a global coarse problem so as to accelerate the convergence of an iterative solution of the global interface problem. The interface transmission conditions are formulated as either a Dirichlet-Neumann or a Robin-Robin mapping by using either one-sided or two-sided Lagrange multipliers. The second scheme outperforms the first at high frequencies because the Robin-type boundary condition does not support resonant modes with a real-valued frequency even when the electrical size of the subdomain increases beyond the cut-off frequency of the lowest-order resonant mode of the subdomain. Therefore, no slowdown in the convergence of the iterative solution of the interface equation occurs. Another nonoverlapping DDM called the FETI-like method, which is based on cement elements, has also made great progress during the past six years. Similar to the FETI-DP method, this algorithm requires the Robin-type continuity across the subdomain interfaces to preserve the fast convergence. However, such a continuity condition is only enforced in a weak sense, which makes the cement FETI-like method capable of dealing with nonconformal interface meshes in a straightforward manner. In some engineering applications, it is necessary to first break the entire computational domain into smaller subdomains and then mesh each subdomain individually if the problem is very large. In such a case, it is difficult to avoid nonconformal meshes at the interface between two neighboring subdomains. In order to further enhance the capability of the FETI-DP method to deal with nonconformal meshes, we proposed two nonconforming FETI-DP methods in our recent work. They are referred to as the Lagrange-multiplier (LM)-based FETI-DP method and the cement-element (CE)-based FETI-DP method, respectively. As the name indicates, the first one is to extend the conformal FETI-DP method to nonconformal cases, while the second one evolves from the cement FETI-like method by introducing a global corner related coarse problem. The beauty of LM-based FETI-DP is to avoid the use of the interface magnetic field and generate a system with only the electric field and Lagrange multiplier. By doing so, the subdomain matrix in the LM-based FETI-DP method has a Because of the smaller subdomain matrices, the LM-based FETI-DP method has faster forward and backward substitution in the direct solution of the subdomain matrix, which are required in the iterative solution of the global system matrix as well as in the solution of the fields in the local subdomains. Also, numerical examples show that these two methods converge with almost the same number of steps due to the usage of the Robin-type transmission condition and the extraction of the global corner unknowns.