Algebraic properties of BEM-FEM coupling with Whitney elements

Purpose - To introduce a Whitney-element based coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM); to discuss the algebraic properties of the resulting system and propose solver strategies. Design/methodology/approach - The FEM is interpreted in the framework of the theory of discrete electromagnetism (DEM). The BEM formulation is given in a DEM-compatible notation. This allows for a physical interpretation of the algebraic properties of the resulting BEM-FEM system matrix. To these ends we give a concise introduction to the mathematical concepts of DEM. Findings - Although the BEM-FEM system matrix is not symmetric, its kernel is equivalent to the kernel of its transpose. This surprising finding allows for the use of two solution techniques: regularization or an adapted GAMS solver. Research limitations/implications - The programming of the proposed techniques is a work in progress. The numerical results to support the presented theory are limited to a small number of test cases. Practical implications - The paper will help to improve the understanding of the topological and geometrical implications in the algebraic structure of the BEM-FEM coupling. Originality/value - Several original concepts are presented: a new interpretation of the FEM boundary term leads to an intuitive understanding of the coupling of BEM and FEM. The adapted GMRES solver allows for an accurate solution of a singular, unsymetric system with a right-hand side that is not in the image of the matrix. The issue of a grid-transfer matrix is briefly mentioned.