Algebraic properties of BEM‐FEM coupling with Whitney elements

Abstract

PurposeTo 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/approachThe 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.FindingsAlthough 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 GMRES solver.Research limitations/implicationsThe 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 implicationsThe paper will help to improve the understanding of the topological and geometrical implications in the algebraic structure of the BEM‐FEM coupling.Originality/valueSeveral 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.