Nonconforming methods for Maxwell source problems and eigenproblems
Abstract:
Partially motivated by the observation that the curl-curl operator behaves differently when it is applied to the divergence-free field and the gradient field in the Hodge decomposition of a function, we introduce the reduced time-harmonic Maxwell (RTHM) equations whose solution is the divergence-free component of the solution to the time-harmonic Maxwell equations. Three schemes are formulated for numerical solving the RTHM system. Two of them use the classical nonconforming finite element approximations, and the other is based on the interior penalty type discontinuous Galerkin methods. To weakly impose the divergence-free condition satisfied by the solutions, the schemes either work with the locally divergence-free trial spaces, or contain a weighted divergence term in the formula. With the properly chosen graded meshes, the optimal error estimates are established which are confirmed by numerical experiments. These schemes and the error estimate results are further extended for solving the reduced curl-curl problems. The discrete operators in these schemes naturally define three Maxwell eigensolvers which are free of spurious eigenmodes and are free of penalty parameters. The analysis for these solvers is closely related to the reduced curl-curl problems and their numerical approximations. Not like those Maxwell eigensolvers based on the full curl-curl problems, the compactness of the involved operator and the uniform error estimates for the source problems greatly simplify the analysis of our proposed eigensolvers.
Applied Mathematics Seminar