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.