hp-Discontinuous Galerkin Finite Element Method for Computational Optics

In computational optics mathematical models and numerical simulations are basic tools to understand the behavior of light in complicated optical devices. The governing laws are well known and given by the Macroscopic Maxwell's equations. In this project a new numerical approach for solv- ing the Maxwell equations in three-dimensional complex domains will be developed, namely an hp-discontinuous Galerkin nite element method (hp- DGFEM). The hp-DGFEM uses locally rened meshes (h-renement) and polynomial approximations of varying degree in each element (p-renement). Since we are considering completely discontinuous nite element spaces, we can easily deal with elements of various shapes and order, and the method is ideally suited for hp-adaptivity. The hp-adaptivity is benecial when one has to deal with local singularities and rapidly changing or discontinuous material properties. A well designed hp-nite element discretization is ca- pable of achieving exponential convergence. Also, hp-DGFEM can easily deal with complex domains using unstructured meshes and the method can preserve accuracy on highly irregular meshes. The project presently is focusing on the design of elements which satisfy the div-curl constraints and on designing a discontinuous Galerkin formulation for the time-dependent Maxwell equations.