Discontinuous Galerkin methods for time-harmonic Maxwell's equations

3 June 2004
Prof Paul Houston
In recent years, there has been considerable interest, especially in the context of fluid-dynamics, in nonconforming finite element methods that are based on discontinuous piecewise polynomial approximation spaces; such approaches are referred to as discontinuous Galerkin (DG) methods. The main advantages of these methods lie in their conservation properties, their ability to treat a wide range of problems within the same unified framework, and their great flexibility in the mesh-design. Indeed, DG methods can easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees. Moreover, orthogonal bases can easily be constructed which lead to diagonal mass matrices; this is particularly advantageous in unsteady problems. Finally, in combination with block-type preconditioners, DG methods can easily be parallelized. \\ \\ In this talk, we introduce DG discretizations of mixed field and potential-based formulations of eddy current problems in the time-harmonic regime. For the electric field formulation, the divergence-free constraint within non-conductive regions is imposed by means of a Lagrange multiplier. This allows for the correct capturing of edge and corner singularities in polyhedral domains; in contrast, additional Sobolev regularity must be assumed in the DG formulation, and their conforming counterparts, when regularization techniques are employed. In particular, we present a mixed method involving discontinuous $P^\ell-P^\ell$ elements, which includes a normal jump stabilization term, and a non-stabilized variant employing discontinuous $P^\ell-P^{\ell+1}$ elements.The first formulation delivers optimal convergence rates for the vector-valued unknowns in a suitable energy norm, while the second (non-stabilized) formulation is designed to yield optimal convergence rates in both the $L^2$--norm, as well as in a suitable energy norm. For this latter method, we also develop the {\em a posteriori} error estimation of the mixed DG approximation of the Maxwell operator. Indeed, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of the energy norm, are derived. \\ \\ Numerical examples illustrating the performance of the proposed methods will be presented; here, both conforming and non-conforming (irregular) meshes will be employed. Our theoretical and numerical results indicate that the proposed DG methods provide promising alternatives to standard conforming schemes based on edge finite elements.
  • Computational Mathematics and Applications Seminar