In this paper we consider the finite element 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 finite element 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 finite element interpolations of arbitrary order. Stability and convergence results are provided for the two finite element formulations considered.

翻译：本文研究了Maxwell问题的有限元逼近，并分析了利用Nitsche方法以弱形式施加本质边界条件的过程。为避免问题的不定性，在原方程组中增加了一个标量场的梯度项，该标量场允许施加磁感应的零散度条件，即使该标量场的精确解为零。我们考虑了两种有限元逼近：一种是假定逼近空间满足适当的inf-sup条件，从而使标准Galerkin方法保持稳定；另一种是增广且稳定的逼近，允许使用任意阶的有限元插值。针对这两种有限元格式，给出了稳定性和收敛性分析结果。