Multilevel Characteristic Basis Finite-Element Method (ML-CBFEM)—An Efficient Version of a Domain Decomposition Algorithm for Large-Scale Electromagnetic Problems

We introduce a memory-efficient version of the Characteristic Basis Finite-Element Method (CBFEM), which combines the domain decomposition with the use of characteristic basis functions (CBFs) that are tailored for each individual subdomain. Although the conventional CBFEM is inherently an efficient approach, the final number of unknowns is primarily determined by the size (or the number) of the subdomains. The larger the size of the subdomains, or fewer the number, the less is the final number of unknowns. However, if we employ “large” subdomains, it is more difficult to generate CBFs for each subdomain due to the memory bottleneck in utilizing direct solution techniques employed to generate the CBFs. In the proposed multilevel approach, referred to herein as the Multilevel CBFEM (ML-CBFEM), we first decompose the computational domain into several “smaller” subdomains, and generate the CBFs for each subdomain in a conventional manner. Then, these bases are combined in a multilevel fashion to derive the CBFs for larger subdomains. In each level, the CBFs are created by using the bases in the lower level. This approach, also called “nested” CBFEM, leads to a considerable reduction in the matrix size and memory, and thus, makes use of direct solvers efficiently.

I. Introduction

Currently, there exists great interest in developing state-of-the-art numerical techniques for efficient simulation of multiscale electromagnetic problems involving electrically large and/or geometrically complex objects. Domain decomposition technique is an efficient way of handling problems of this type, and, over the last decade, a number of domain decomposition schemes have been proposed in the area of electromagnetics [1]–[8]. The characteristic basis function method (CBFM), which has been initially introduced in the context of method of moments (MoM) [9], is a relatively new domain decomposition approach for solving large-scale electromagnetic problems that utilizes high-level basis functions, called the characteristic basis functions (CBFs). In this approach, the CBFs are generated for each subdomain by considering the physics of the problem. They are used to transform the original matrix into a “smaller” one, called the reduced matrix. Another CBF-based approach has been utilized for the first time in FEM, and has been named characteristic basis finite-element method (CBFEM). This method, whose details of its implementation are considerably different from those followed in the previous MoM-based approaches, has been applied to a number of representative problems in both the quasi-static [10] and time-harmonic regimes [11], [12]. The CBFs have been generated in the above works by using point charges and dipole-type sources, for quasi-static and time-harmonic problems, respectively. Recently, another type of CBFEM has been proposed by using the principles of physical optics (PO) for the generation of the CBFs [13].