We analyze the conforming approximation of the time-harmonic Maxwell's equations using N\'ed\'elec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed.

翻译：我们分析了使用Nédélec（边）有限元对时谐麦克斯韦方程组的协调逼近。我们证明该逼近是渐近最优的，即能量范数下的逼近误差被最佳逼近误差乘以一个随着网格细化及/或多项式次数增加而趋近于1的常数所界定。此外，在相同的网格及/或多项式次数条件下，我们建立了离散inf-sup稳定性，其常数至多对应于连续问题常数的两倍。我们的证明仅需要精确解具有最低正则性假设，因此允许一般区域、材料系数和右端项。