Coupling of the ultra-weak variational formulation and an integral representation using a fast multipole method in electromagnetism
The ultra-weak variational formulation (UWVF) is a volume-based numerical method for solving the time-harmonic Maxwell system on a bounded domain developed by Despres and Cessenat. It uses local plane wave solutions on a finite element mesh to approximate the field. By varying the number of plane wave basis functions from element to element the UWVF can discretize the electromagnetic field with a coarser volume mesh in comparison to more classical methods like low order finite elements or finite differences. However, to approximate scattering on an unbounded domain, the UWVF requires an artificial boundary \(\Gamma_{\text{ext}}\), sufficiently far from the obstacle. A simple absorbing boundary condition on \(\Gamma_{\text{ext}}\), (as used in the original UWVF) implies a large domain around the obstacle and so a large number of degrees of freedom. In this paper the authors suggest using an integral representation of the unknown field on \(\Gamma_{\text{extt}}\) due to Hazard and Lenoir. In particular, their idea is to use an integral representation of the unknown on the artificial boundary \(\Gamma_{\text{ext}}\) thanks to the unknown field values on a third boundary \(\Sigma\) taken closer to the boundary \(\Gamma_{\text{int}}\). This couples the degrees of freedom on \(\Sigma\) to those on \(\Gamma_{\text{ext}}\). The main constraint is that the domain between \(\Sigma\) and \(\Gamma_{\text{ext}}\) be homogeneous (i.e. the background medium). The artificial boundary \(\Gamma_{\text{ext}}\) can then be taken very close to the boundary of the obstacle. However, this method requires the evaluation of integral operators which are expensive by direct means. They show that the use of the integral representation does not greatly increase the cost of the UWVF if the integral calculation is performed using a fast multipole method (FMM). Indeed, for a given accuracy, the numerical complexity of the new algorithm is better than the complexity of the standard UWVF with a simple absorbing boundary condition on \(\Gamma_{\text{ext}}\). The article gives a presentation of the UWVF and describes the use of the integral representation within the UWVF, with results on the complexity of the new algorithm. The last section presents encouraging numerical results obtained using a 1-level FMM.
scientific article; zbMATH DE number 5161213