Finite Elements in Action

Abstract The finite element method (FEM) delivers very high accuracy through the use of Hermite interpolation and the discretization of the action integral. A variation of the nodal interpolation parameters yields the discretized Schrödinger and the Maxwell equations. Boundary conditions of the Dirichlet, Neumann, and Cauchy types are easily implemented in this framework. Scattering theory in 1D, 2D, and 3D is shown to be represented by an action integral with mixed boundary conditions, thereby allowing variational techniques with finite elements to be employed with advantage. Through the use of totally absorbing regions enclosing the scattering center, together with a source term for the incident wave, we reduce the radiation BC to a Dirichlet BC over a finite domain. This yields evanescent wave coefficients as well as the usual propagating solutions with high accuracy. We demonstrate the efficacy of Hermite finite elements for calculating electromagnetic fields in waveguides, photonic crystals, and in cavities with re-entrant peripheries. Singular field gradients are tamed using dimensional continuation and the fields mapped in 3D. A strategy for eliminating spurious modes is developed. The presence of accidental degeneracy in energy for a quantum mechanical particle in a cube, and for photons in a cubic cavity, and their removal through perturbations, is shown through modeling and employing group theory. The shape functions for symmetric finite elements themselves are also determined using group theory. With interface smoothing using Fermi functions, we remove the need for “jump conditions” at interfaces. The combination of variational techniques together with the principle of stationary action is shown to be a very powerful combination when combined with the finite element method for solving physical problems.