The principal computational electromagnetics techniques for solving antenna problems are reviewed. An introduction is given on a historical review of how antenna problems were solved in the past. The call for precise solutions calls for the use of numerical methods as found in computational electromagnetics. A brief introduction on differential equation solutions and integral solutions is given. The Green's function concept is introduced to facilitate the formulation of integral equations. Numerical methods and fast algorithms to solve these equations are discussed.Then an overview of how electromagnetic theory relates to circuit theory is presented. Then the concept of partial element equivalence circuit is introduced to facilitate solutions to more complex problems. In antenna technology, one invariably has to have a good combined understanding of the wave theory and circuit theory.Next, the discussion on the computation of electromagnetic solutions in the "twilight zone" where circuit theory meets wave theory was presented. Solutions valid for the wave physics regime often become unstable facing low-frequency catastrophe when the frequency is low.Due to advances in nanofabrication technology, antennas can be made in the optical frequency regime. But their full understanding requires the full solutions of Maxwell's equations. Also, many models, such as perfect electric conductors, which are valid at microwave frequency, are not valid at optical frequency. Hence, many antenna concepts need rethinking in the optical regime.Next, an emerging area of the use of eigenanalysis methods for antenna design is discussed. This can be the characteristic mode analysis or the natural mode analysis. These analysis methods offer new physical insight not possible by conventional numerical methods.Then the discussion on the use of the domain decomposition method to solve highly complex and multi-scale antenna structures is given. Antennas, due to the need to interface with the circuit theory, often have structures ranging from a fraction of a wavelength to a tiny fraction of a wavelength. This poses a new computational challenge that can be overcome by the domain decomposition method.Many antenna designs in the high-frequency regime or the ray optics regime are guided by ray physics and the adjoining mathematics. These mathematical techniques are often highly complex due to the rich physics that come with ray optics. The discussion on the use of these new mathematical techniques to reduce computational workload and offering new physical insight is given.A conclusion section is given to summarize this chapter and allude to future directions.

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This paper reviews various methods to solve multiscale problems ranging from low-frequency methods to very high-frequency methods.

To obtain second order accuracy using the first order data for the capacitance parameter extraction, a set of new generalized variational formulas are presented in this paper. They are complete since they are valid not only for self-capacitances, but also for mutual capacitances. A more general variational formula is derived to account for the asymmetrical elastance matrix case. By these novel formulas the computational accuracy can be significantly improved compared to the conventional direct capacitance extraction method.

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