Transfinite elements: A highly efficient procedure for modeling open field problems

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Abstract
This paper presents a new approach for modeling open boundary conditions in electric and magnetic field problems that is considerably simpler and more efficient than the existing alternatives. The procedure is called the transfinite element method because it employs analytic basis functions to transfer the exterior boundary condition over an infinite domain to a finite-sized numerical solution region. In the transfinite element method, the problem region is divided into an interior region Ωi and an exterior region Ωe by a circular boundary Γ. Since the exterior region involves only a circular boundary, it may be solved analytically using the separation of variables procedure. Substituting this solution into the variational principle for the electric or magnetic field, minimizing it, and imposing continuity between the finite element solution in the interior region and the analytical solution in the exterior region results in a real, symmetric matrix equation that is solved for the field.

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