An analysis of Nitsche’s prescription of Dirichlet conditions for the conforming finite element approximation of Maxwell’s problem

In this paper we consider the conforming finite element (FE) approximation of Maxwell’s problem and analyse the prescription of essential boundary conditions in a weak sense using Nitsche’s method. To avoid indefiniteness of the problem, the original equations are augmented with the gradient of a scalar field that allows one to impose the zero divergence of the magnetic induction, even if the exact solution for this scalar field is zero. Two FE approximations are considered, namely, one in which the approximation spaces are assumed to satisfy the appropriate inf-sup condition that render the standard Galerkin method stable, and another augmented and stabilised one that permits the use of FE interpolations of arbitrary order. Stability and convergence results are provided for the two FE formulations considered.