We consider time-harmonic Maxwell's equations set in an heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in $L^2$, we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order N\'ed\'elec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell's equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the right-hand side only exhibits $L^2$ regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation, and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmoltz equation and showing the interest of high-order methods.
翻译:我们认为时间- 调和 Maxwell 的方程式设置在一个具有完美运行边界条件的多元介质中。 鉴于无差异的右手边以2美元为单位, 我们提供一种频率的可理解性估计值, 测量相应的解决方案与其以高序 N\'ed\'elec 限定元素最接近值之间的差异。 这种近似性估计值在对 Maxwell 方程式的有限元素分解值的先验和后验错误分析中都至关重要, 但衍生值并非微不足道。 事实上, 很难利用高序的多义方程式, 因为右侧只显示2美元为常规值。 我们遵循先前获得的结果, 更简单的设定 selmholtz 方程式, 并提出解决方案的定期分割 。 反过来, 这种分裂产生惊人的相容性估计值, 概括了已知的 scalar Helmoltz 方程式结果, 并展示了高序方法的兴趣 。