  A hybrid surface integral equation partial differential equation (SIE-PDE)
formulation without the boundary condition requirement is proposed to solve the
transverse magnetic (TM) electromagnetic problems. In the proposed formulation,
the computational domain is decomposed into two overlapping domains: the SIE
and PDE domains. In the SIE domain, complex structures with piecewise
homogeneous media, e.g., highly conductive media, are included. An equivalent
model for those structures is constructed by replacing them with the background
medium and introducing a surface equivalent electric current density on an
enclosed boundary to represent their electromagnetic effects. The remaining
computational domain and homogeneous background medium replaced domain consist
of the PDE domain, in which inhomogeneous or non-isotropic media are included.
Through combining the surface equivalent electric current density and the
inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. It
requires no boundary conditions, and is mathematically equivalent to the
original physical model. Through careful construction of basis functions to
expand electric fields and the equivalent current density, the discretized
formulation is made compatible with the SIE and PDE domain interface. The
accuracy and efficiency are validated through two numerical examples. Results
show that the proposed SIE-PDE formulation can obtain accurate results, and
significant performance improvements in terms of CPU time and memory
consumption compared with the FEM are achieved.
