A Rigorous Code Verification Process of the Domain Decomposition Method in a Finite Element Method for Electromagnetics

The finite element method (FEM) has benefited recently from the introduction of the domain decomposition method (DDM), especially to tackle large-scale problems. Many versions of DDM may be found in the literature, and almost none of them has used a rigorous approach to show the accuracy of the formulation. Here, we suggest the use of the method of manufactured solutions (MMS) to introduce DDM into an FEM code. This approach also allows us to debug the demanding coding process of the method. First, we introduce the mandatory operators used for DDM to construct second-order absorbing boundary conditions (ABCs). Then, we use these results to show the correct implementation of DDM with basis functions up to fourth order for tetrahedra, triangular prisms, and hexahedra, obtaining the convergence rates predicted by the classic FEM theory. Finally, we illustrate how to use MMS to test different formulations, assessing the effect of using different spaces and orders for the arising ancillary variables.