A Finite Element Method Approach to the Finite-Difference Time-Domain Scheme with Cut Cells

Abstract

In this article, a finite element method approach to the finite-difference time-domain scheme with cut cells for Maxwell's equations in two dimensions is presented and tested. The method is based on a structured Cartesian grid of square elements, where cut cells are introduced for curved or oblique boundaries. The field is approximated by piecewise bilinear basis functions, and it is allowed to be discontinuous at interfaces modeled by cut cells, where continuity is enforced in the weak sense by means of Nitsche's method. Trapezoidal quadrature on the uncut elements recovers the finite-difference approximation that features a diagonal mass matrix. The time-domain formulation of this method is based on a hybrid explicit-implicit time-stepping method, which reduces to the leap-frog scheme for the uncut elements and the unconditionally stable Newmark method for the cut elements. The global time step is bounded by the Courant condition of the finite-difference time-domain scheme. For problems with regular solutions, the method yields a discretization error that is proportional to the square of the cell size. The method is tested on square cavities in two dimensions, where the first test problem involves homogeneous material parameters, and the second problem features a dielectric region in a vacuum cavity.