In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell’s equations.

在本文中，我们研究了具有对比扩散率的扩散模型问题的近似，适用于各种非一致性近似方法。本质上的困难是精确解的 Sobolev 平滑度指数可能仅略大于 1。通过在网格面上赋予精确解的法向导数一个弱意义来处理缺乏平滑度。我们推导出稳健和准最优的误差估计。准最优性意味着近似误差受到离散试验空间中最佳近似误差的限制，直到一个通用常数，而稳健性意味着通用常数与扩散率对比无关。误差估计使用与网格相关的范数，该范数在离散水平上是等效的，