A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the 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 through replacing them by 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. Unlike other hybrid formulations, where the transmission condition is usually used, no boundary conditions are required in the proposed SIE-PDE formulation, and it 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 compatible on the interface of the SIE and PDE domain. Finally, its accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results including both near and far fields, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

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A nonconformal domain decomposition method based on the hybrid surface integral equation partial differential equation (SIE-PDE) formulation is proposed to solve the transverse magnetic electromagnetic problems. In the hybrid SIE-PDE formulation, an equivalent model with only the electric current density is first constructed, and then is embedded into the inhomogeneous Helmholtz equation as an excitation. A connection matrix, which couples the interfaces of the SIE and PDE domains, is carefully designed to support nonconformal meshes. Since meshes in each domain are independently generated, it is much more efficient and flexible to model multiscale and complex structures compared with the original hybrid SIE-PDE formulation with conformal meshes. The proposed formulation is efficient, flexible and easy to implement. Its accuracy, efficiency and flexibility are validated by three numerical examples.