Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Transparent Boundary Conditions

\(A\lesssim B\) is shorthand for \(A \le C B\) for some \(C>0\) that is independent of the wavenumber k, the mesh size h, the polynomial degree p, as well as functions appearing in A and B.

The condition \(k\ge 1\) can be replaced by \(k\ge k_{0}>0\). Our estimates remain valid for all choices of \(k_{0}>0\). The constants in the estimates are uniform for all \(k\ge k_{0}\), while they depend continuously on \(k_{0}\) and, possibly, become large as \(k_{0}\rightarrow 0\). We use (2.2) simply to reduce technicalities.

The function spaces appearing in these statements will be introduced in Sect. 2.3.

This follows by representing \(T_{k}\) by trace operators and boundary/volume potentials for the electric Maxwell equation as, e.g., explained in [11], and by applying the rules for computing the adjoint of composite operators.

We write \({\tilde{\eta }}_{\ell }\) for an approximation property which involves a solution operator and \(\eta _{\ell }\) for a “pure” approximation property for a given space/set of functions.

For the last relation, we have estimated \(\left\| \cdot \right\| _{{\text {curl}},\varOmega ,1} \le \left\| \cdot \right\| _{{\text {curl}},\varOmega ,k}\) in (5.30) (using (2.2)) to simplify technicalities.

The third condition is not essential but leads to a significant simplification as the ensuing (5.37) effects a decoupling of the elliptic system (5.38) into three scalar problems at 0.

In [40, Def. 5.3, Thm. B.4] the element-by-element construction of the polynomial approximation on the reference element only fixes \(\varPi _{p}\) on \(\partial \widehat{K}\). The operator \(\varPi _{p}\) is fully determined by adding a final minimization step to fix the interior degrees of freedom on the reference element.

There is a sign error in the second last relation on [30, p.271].

the factor 3 in \(3C_{I}\) is due to the summation over i and likely suboptimal.