Coupling of the ultra-weak variational formulation and an integral representation using a fast multipole method in electromagnetism

Many different methods have been developed for the solution of the time-harmonic Maxwell equations in exterior domains at high frequency. Volume-based methods have the drawback of needing an artificial boundary far from the obstacle. Integral formulations enable one to avoid this difficulty by solving a problem on the surface of the obstacle. However, integral operators imply dense systems with bad condition numbers. The ultra-weak variational formulation (UWVF) is a volume-based method using plane wave basis functions that allows the use of a coarser mesh in comparison with more classical low order finite element methods. However, the UWVF still involves the problem of the artificial boundary. In this paper, we suggest the use of an integral representation of the unknown field to obtain an exact artificial boundary condition. In this way the distance between the obstacle and the artificial boundary can be reduced. The use of the fast multipole method ensures a low cost for the calculation of various integral operators used in the representation. In this paper we describe the combined algorithm, demonstrate its accuracy on a model problem and discuss the complexity of the algorithm.