Applications of differential forms to boundary integral equations

In this paper we discuss the application of differential forms to integral equations arising in the study of electromagnetic wave propagation. The usual Stratton-Chu integral equations are derived in terms of differential forms and corresponding Galerkin formulations are constructed. All numerical schemes require the specification of basis functions and the use of differential forms provides a very general method for the construction of arbitrary order basis functions on curvilinear geometries. It is noted that the lowest order approximations on flat geometries reduce to forms essential equivalent to the standard Rao-Wilton-Glisson functions. The effect on accuracy is investigated for electric field integral equation and magnetic field integral equation formulations for a range of bases. Hierarchical classes of functions are also developed, as are transition elements useful in p-adaptive schemes where variable orders of approximation are sought.