Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions
DOI: 10.1007/s10208-023-09626-7
© The Author(s) 2023
Received: 17 January 2022
Accepted: 11 May 2023
Published: 14 November 2023
Abstract
The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that (a) kh/p is sufficient small and (b) \(p/\ln k\) is bounded from below.
Keywords
Maxwell’s equations Time-harmonic High-frequency Wavenumber explicit hp-FEM Quasi-optimalityMathematics Subject Classification
35J05 65N12 65N301 Introduction
The time-harmonic Maxwell equations at high wavenumber k are a fundamental component of high-frequency computational electromagnetics. Computationally, these equations are challenging for several reasons. The solutions are highly oscillatory so that fine discretizations are necessary and correspondingly large computational resources are required. While conditions to resolve the oscillatory nature of the solution appear unavoidable, even more stringent conditions on the discretizations have to be imposed for stability reasons: In many numerical methods based on the variational formulation of Maxwell’s equations, the gap between the actual error and the best approximation error widens as the wavenumber k becomes large. This “pollution effect” is a manifestation of a lack of coercivity of the problem, as is typical in time-harmonic wave propagation problems. Mathematically understanding this “pollution effect” in terms of the wavenumber k and the discretization parameters for the model problem (1.1) is the purpose of the present work.
The “pollution effect”, i.e., the fact that discretizations of time-harmonic wave propagation problems are prone to dispersion errors, is probably best studied for the Helmholtz equation at large wavenumbers. The beneficial effect of using high order methods was numerically observed very early and substantiated for translation-invariant meshes [1, 2]; a rigorous mathematical analysis for unstructured meshes was developed in the last decade only in [17, 37, 38]. These works analyze high order FEM (hp-FEM) for the Helmholtz equation in a Gårding setting using duality techniques. This technique, often called “Schatz argument”, crucially hinges on the regularity of the dual problem, which is again a Helmholtz problem. The key new insight of the line of work [17, 37, 38] is a refined wavenumber-explicit regularity theory for Helmholtz problems that takes the following form (“regularity by decomposition”): given data, the solution u is written as \(u_{H^{2}}+u_{{\mathcal {A}}}\) where \(u_{H^{2}}\) has the regularity expected of elliptic problems and is controlled in terms of the data with constants independent of k. The part \(u_{{\mathcal {A}}}\) is a (piecewise) analytic function whose regularity is described explicitly in terms of k. Employing “regularity by decomposition” for the analysis of discretizations has been successfully applied to other Helmholtz problems and discretizations such DG methods [31], BEM [27], FEM-BEM coupling [29], and heterogeneous Helmholtz problems [4, 9, 25, 26].
Compared to the scalar Helmholtz case, where the compact embedding \(H^1 \subset L^2\) underlies the success of the duality argument, the convergence analysis of discretizations of Maxwell’s equations is hampered by the fact that the embedding \({\textbf{H}}({\text {curl}}) \subset {\textbf{L}}^2\) is not compact so that a duality argument is not immediately applicable. This issue arises even in the context of convergence analyses that are not explicit in the wavenumber k. An analysis can be based on the observation that \({\textbf{H}}({\text {curl}}) \cap {\textbf{H}}({\text {div}})\) endowed with appropriate boundary conditions is compactly embedded in \({\textbf{L}}^2\). This approach, which is structurally described in [39, Sec. 1.2], involves as a first ingredient the ability to decompose discrete functions into gradient parts and (discrete) solenoidal parts in two ways, namely, on the continuous level and the discrete level. The solenoidal part of the decomposition on the continuous level is in \({\textbf{H}}({\text {curl}}) \cap {\textbf{H}}({\text {div}})\) and admits a duality argument. Galerkin orthogonalities are invoked to then reduce the analysis to that of the difference between the solenoidal parts of the continuous and the discrete level. For the analysis of this difference, a second ingredient is vital, namely, special interpolation operators with a commuting diagram property. These two ingredients underlie many duality arguments for Maxwell problems in the literature, see, e.g., [41, Sec. 7.2], [8, 10, 16, 56] and references therein. The present work follows [41, Sec. 7.2] and the path outlined in [39, Sec. 1.1–1.3].
At the heart of the k-explicit convergence analysis for (1.1) is a k-explicit regularity theory for the above mentioned dual problem. Similarly to the Helmholtz case discussed above, it takes the form of a “regularity by decomposition” (Theorem 7.3). Such a regularity theory was developed for Maxwell’s equations in full space in the recent paper [39], where the decomposition is directly accessible in terms of the Newton potential and layer potentials. For the present bounded domain case, however, an explicit construction of the decomposition is not available, and the iterative construction as in the Helmholtz case of [38] has to be brought to bear. For this, a significant complication in the Maxwell case compared to the Helmholtz case arises from the requirement that the frequency filters used in the construction be such that they produce solenoidal fields if the argument is solenoidal.
While our wavenumber-explicit regularity result Theorem 7.3 underlies our proof of quasioptimal convergence of the high order Galerkin method (cf. Theorem 9.7), it also proves useful for wavenumber-explicit interpolation error estimates as worked out in Corollary 9.8.
The present paper analyzes an \({{\textbf{H}}}({\text {curl}})\)-conforming discretization based on high order Nédélec elements. Various other high order methods for Maxwell’s equations that are explicit in the wavenumber can be found in the literature. Closest to our work are [11, 45]. The work [45] studies the same problem (1.1) but uses an \({{\textbf{H}}}^{1}\)-based instead of an \({{\textbf{H}}}({\text {curl}} )\)-based variational formulation involving both the electric and the magnetic field. The proof of quasi-optimality in [45] is based on a “regularity by decomposition” technique similar to the present one. [44] studies the same \({{\textbf{H}}}^{1}\)-based variational formulation and \({{\textbf{H}}}^{1} \)-conforming discretizations for (1.1) on certain polyhedral domains and obtains k-explicit conditions on the discretization for quasi-optimality. Key to this is a description of the solution regularity in [44] in terms of corner and edge singularities. The work [11] studies fixed (but arbitrary) order \({{\textbf{H}}}({\text {curl}})\)-conforming discretizations of heterogeneous Maxwell problems and shows a similar quasi-optimality result by generalizing the corresponding Helmholtz result [9]; the restriction to finite order methods compared to the present work appears to be due to the difference in which the decomposition of solutions of Maxwell problems is obtained. High order Discontinuous Galerkin (DG) and Hybridizable DG (HDG) methods for (1.1) have been presented in [18] and [28] together with a stability analysis that is explicit in h, k, and p. A dispersion analysis of high order methods on tensor-product meshes is given in [2].
The outline of the paper is as follows. Section 2 introduces the notation and tools such as regular decompositions (see Sect. 2.4) that are indispensable for the analysis of Maxwell problems. Section 3 (Theorem 3.7) shows that the solution of (1.1) depends only polynomially on the wavenumber k. This stability result is obtained using layer potential techniques in the spirit of earlier work [17, Thm. 2.4] for the analogous Helmholtz equation. While earlier stability estimates for (1.1) in [18, 23, 55], and [44, Thm. 5.2 ] are obtained by judicious choices of test functions and rely on star-shapedness of the geometry, Theorem 3.7 does not require star-shapedness. It is worth mentioning that at least in the analogous case of the Helmholtz equation, alternatives to the use of suitable test functions or layer potential exist, which can lead to better k-dependencies; we refer to [51] for results and a discussion. Section 4 analyzes a “sign definite” Maxwell problem and presents k-explicit regularity assertions for it (Theorem 4.3). The motivation for studying this particular boundary value problem is that, since the principal parts of our sign-definite Maxwell operator and that of (1.1) coincide, a contraction argument can be brought to bear in the proof of Theorem 7.3. A similar technique has recently been used for heterogeneous Helmholtz problems in [4]. Section 5 collects k-explicit regularity assertions for (1.1) (Lemma 5.1 for finite regularity data and Theorem 5.2 for analytic data). The contraction argument in the proof of Theorem 7.3 relies on certain frequency splitting operators (both in the volume and on the boundary), which are provided in Sect. 6. Section 7 presents the main analytical result, Theorem 7.3, where the solution of (1.1) with finite regularity data \({\textbf{f}}\), \({\textbf{g}}\) is decomposed into a part with finite regularity but k-uniform bounds, a gradient field, and an analytic part. Section 8 presents the discretization of (1.1) based on high order Nédélec elements and presents hp-approximation operators that map into Nédélec spaces. These operators are the same ones as used in [39] but we work out their approximation properties on the skeleton of the mesh since stronger approximation properties on the boundary \(\partial \varOmega \) are required in the present case of impedance boundary conditions. Section 9 shows quasi-optimality (Theorem 9.7) under the scale resolution condition (1.2). Section 10 concludes the paper with numerical results.
2 Setting
2.1 Geometric Setting and Sobolev Spaces on Lipschitz Domains
Let \(\varOmega \subset {\mathbb {R}}^{3}\) be a bounded Lipschitz domain which we assume throughout the paper to have a simply connected and sufficiently smooth boundary \(\varGamma :=\partial \varOmega \); if less regularity is required, we will specify this. We flag already at this point that the main quasi-optimal convergence result, Theorem 9.7 will require analyticity of \(\varGamma \). The outward unit normal vector field is denoted by \({\textbf{n}}:\varGamma \rightarrow {\mathbb {S}}_{2}\).
Definition 2.1
2.2 Sobolev Spaces on a Sufficiently Smooth Surface \(\varGamma \)
The Sobolev spaces on the boundary \(\varGamma \) are denoted by \(H^{s}( \varGamma ) \) for scalar-valued functions and by \({\textbf{H}}^{s}( \varGamma ) \) for vector-valued functions with norms \(\left\| \cdot \right\| _{H^{s}( \varGamma ) }\), \(\left\| \cdot \right\| _{{\textbf{H}}^{s}( \varGamma ) }\) (see, e.g., [30, p. 98]). Note that the range of s for which \(H^{s}( \varGamma ) \) is defined may be limited, depending on the global smoothness of the surface \(\varGamma \); for Lipschitz surfaces, s can be chosen in the range \(\left[ 0,1\right] \). For \(s<0\), the space \(H^{s}( \varGamma ) \) is the dual of \(H^{-s}( \varGamma ) \).
Lemma 2.2
2.3 Trace Operators and Energy Spaces for Maxwell’s Equations
Proposition 2.3
Remark 2.4
For gradient fields \(\nabla \varphi \) we have \((\nabla \varphi )_{T}^{{\text {curl}}}=0\) and \((\nabla \varphi )_{T}^{\nabla }=\nabla _{\varGamma }\varphi \). \(\square \)
Definition 2.5
2.4 Regular Decompositions
We will rely on various decompositions of functions into regular parts and gradient parts. The decompositions may not be orthogonal but must be stable. We refer to [20, §4.4] and the bibliographic notes therein for some early contributions. Many variants have been introduced since then, and the results in this section are essentially taken from the literature: Lemma 2.6 is a consequence of [15, Thm. 4.6]; Lemma 2.7 relies on [41, Thm. 3.38] and Lemma 2.6; Lemma 2.8 is based on [13] while closely related results can be found in [3]. Finally, Lemma 2.9 is a consequence of [50, Thm. 4.2(2)] and [13]. For newer overview articles, we refer to, e.g., [21, 24]. The following Lemma 2.6 collects a key result from the seminal paper [15]. The operator \({\textbf{R}}_{2}\), which is essentially a right inverse of the curl operator, will frequently be employed in the present paper.
Lemma 2.6
Proof
Lemma 2.7
- (i)There is \(C>0\) such that for every \({\textbf{u}}\in {\textbf{X}}\) there is a decomposition \({\textbf{u}} =\nabla \varphi +{\textbf{z}}\) with$$\begin{aligned} {\text {div}}\,{\textbf{z}}=0,\quad \left\| {\textbf{z}}\right\| _{{\textbf{H}}^{1}(\varOmega )}\le C\left\| {\text {curl}}\,{\textbf{u}} \right\| ,\quad \left\| \varphi \right\| _{H^{1}(\varOmega )}\le C\left\| {\textbf{u}}\right\| _{{\textbf{H}}\left( {\text {curl}},\varOmega \right) }. \nonumber \\ \end{aligned}$$(2.28)
- (ii)Let \(m\in {\mathbb {Z}}\). For each \({\textbf{u}}\in {\textbf{H}}^{m}( {\text {curl}},\varOmega ) \) there is a splitting independent of m of the form \({\textbf{u}}=\nabla \varphi +{\textbf{z}}\) with \(\varphi \in H^{m+1}( \varOmega ) \), \({\textbf{z}}\in {\textbf{H}}^{m+1}( \varOmega ) \) satisfying$$\begin{aligned} \left\| {\textbf{z}}\right\| _{{\textbf{H}}^{m+1}(\varOmega )}&\le C\left\| {\textbf{u}}\right\| _{{\textbf{H}}^{m}( {\text {curl}} ,\varOmega ) }\quad \text {and}\quad \end{aligned}$$(2.29a)$$\begin{aligned} \left\| {\textbf{z}}\right\| _{{\textbf{H}}^{m}(\varOmega )}+\left\| \varphi \right\| _{H^{m+1}(\varOmega )}&\le C\left\| {\textbf{u}}\right\| _{{\textbf{H}}^{m}( \varOmega ) }. \end{aligned}$$(2.29b)
- (iii)There is \(C > 0\) depending only on \(\varOmega \) such that each \({\textbf{u}} \in {\textbf{X}} _{{\text {imp}}}\) can be written as \({\textbf{u}} = \nabla \varphi + {\textbf{z}}\) with \(\varphi \in H^{1}_{{\text {imp}}}(\varOmega )\), \({\textbf{z}} \in {\textbf{H}}^{1}(\varOmega )\) and$$\begin{aligned} \Vert \nabla \varphi \Vert _{{\text {imp}},k} + \Vert {\textbf{z}}\Vert _{{\textbf{H}} ^{1}(\varOmega )} + |k| \Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varOmega )} + |k|^{1/2} \Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varGamma )} \le C \Vert {\textbf{u}} \Vert _{{\text {imp}},k}. \end{aligned}$$(2.30)
Proof
Proof of (iii): Multiplying estimate (2.29b) for the decomposition of (ii) and \(m=0\) by \(\left| k\right| \) leads to \(\left| k\right| \Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varOmega )}+\left| k\right| \Vert \nabla \varphi \Vert _{{\textbf{L}}^{2}(\varOmega )}\le C\left| k\right| \Vert {\textbf{u}}\Vert _{{\textbf{L}}^{2}(\varOmega )}\). (2.29a) gives \(\Vert {\textbf{z}} \Vert _{{\textbf{H}}^{1}(\varOmega )}\le C\Vert {\textbf{u}}\Vert _{{\textbf{H}} ({\text {curl}},\varOmega )}\). The multiplicative trace inequality gives \(|k|\Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varGamma )}^{2}\le C\left| k\right| \Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varOmega )}\Vert {\textbf{z}} \Vert _{{\textbf{H}}^{1}(\varOmega )}\le C\left| k\right| ^{2} \Vert {\textbf{z}}\Vert _{{\textbf{L}}^{2}(\varOmega )}^{2}+C\Vert {\textbf{z}} \Vert _{{\textbf{H}}^{1}(\varOmega )}^{2}\). Hence follows \(\Vert {\textbf{z}}\Vert _{{\text {imp}},k}\le C\Vert {\textbf{u}}\Vert _{{\textbf{H}} ({\text {curl}},\varOmega ),k}\le C\Vert {\textbf{u}}\Vert _{{\text {imp}},k}\). The triangle inequality then provides the bound \(\Vert \nabla \varphi \Vert _{{\text {imp}},k}\le \Vert {\textbf{u}}\Vert _{{\text {imp}},k}+\Vert {\textbf{z}}\Vert _{{\text {imp}},k}\le C\Vert {\textbf{u}} \Vert _{{\text {imp}},k}\). \(\square \)
The following result relates the space \({\textbf{H}}({\text {curl}},\varOmega )\cap {\textbf{H}}({\text {div}},\varOmega )\) to classical Sobolev spaces. The statement (2.32) is from [13]; closely related results can be found in [3].
Lemma 2.8
Proof
The following lemma introduces some variants of Helmholtz decompositions.
Lemma 2.9
Proof
2.5 Maxwell’s Equations with Impedance Boundary Conditions
3 Stability Analysis of the Continuous Maxwell Problem
In this section we show that the model problem (2.39) is well-posed and that the norm of the solution operator is \(O(|k|^{\theta })\) for suitable choices of norms and some \(\theta \ge 0 \).
3.1 Well-Posedness
Proposition 3.1
Proof
Step 2: From [19, Thm. 4.8] or the technique developed in [5] it follows that the operator induced by \(A_{k}\) is a compact perturbation of an isomorphism and the Fredholm alternative shows well-posedness of the problem. \(\square \)
3.2 Wavenumber-Explicit Stability Estimates
Proposition 3.1 does not give any insight how the (positive) inf-sup constant \(\gamma _{k}\) depends on the wavenumber k. In this section, we introduce the stability constant \(C_{{\text {stab}}}(k)\) and estimate its dependence on k under certain assumptions.
Definition 3.2
Remark 3.3
For the hp-FEM application below, the term \(\left| k\right| ^{\theta }\) will be mitigated by an exponentially converging approximation term so that any finite value \(\theta \ge 0\) leads to an exponential convergence of the discretization. \(\square \)
Next we present an example5 which shows that in general the exponent \(\theta \) in (3.2) cannot be negative.
Example 3.4
Remark 3.5
In the remaining part of this section, we prove estimate (3.2) for certain classes of domains. The following result removes the assumption in [23] for the right-hand side to be solenoidal.
Proposition 3.6
Let \(\varOmega \subset {\mathbb {R}}^{3}\) be a bounded \(C^{2}\) domain that is star-shaped with respect to a ball. Then, Assumption (3.2) holds with \(\theta =0\).
Proof
Theorem 3.7
Remark 3.8
The analyticity requirement of \(\partial \varOmega \) can be relaxed. It is due to our citing [34], which assumes analyticity. \(\square \)
Proof
We estimate \({\mathcal {S}}_{\varOmega ,k}^{{\text {MW}}}({\textbf{j}},{\textbf{g}}_{T})\) (see (2.41)) for given \(({\textbf{j}},{\textbf{g}}_{T})\in {\textbf{L}}^{2}(\varOmega )\times {\textbf{L}}_{T}^{2}(\varGamma )\).
Lemma 3.9
4 Maxwell’s Equations with the “Good” Sign
4.1 Norms
Lemma 4.1
Proof
Proof of (4.7): The proof is analogous to that of (4.6). \(\square \)
Lemma 4.2
Proof
4.2 The Maxwell Problem with the Good Sign
Theorem 4.3
- (i)
The sesquilinear form \(A_{k}^{+}\) satisfies \({\text {Re}} A_{k}^{+}({\textbf{v}},\sigma {\textbf{v}}) = 2^{-1/2} \Vert {\textbf{v}}\Vert ^{2}_{{\text {imp}},k}\) for all \({\textbf{v}} \in {\textbf{X}}_{{\text {imp}}}\), where \(\sigma =\exp \left( \frac{\pi {\text {i}}\,}{4}{\text {sign}}k\right) \).
- (ii)
The sesquilinear form is continuous: \(|A_{k}^{+}({\textbf{u}},{\textbf{v}})| \le \Vert {\textbf{u}} \Vert _{{\text {imp}},k} \Vert {\textbf{v}}\Vert _{{\text {imp}},k}\) for all \({\textbf{u}}\), \({\textbf{v}} \in {\textbf{X}}_{{\text {imp}}}\).
- (iii)The solution \({\textbf{u}} \in {\textbf{X}}_{{\text {imp}}}\) of (4.13) satisfies$$\begin{aligned} \left\| {\textbf{u}}\right\| _{{\text {imp}},k}&\le C\left( |k|^{-1}\left\| {\textbf{f}}\right\| _{{\textbf{L}}^{2}(\varOmega )}+\left| k\right| ^{-1/2}\left\| {\textbf{g}}_{T}\right\| _{{\textbf{L}} ^{2}(\varGamma )}\right) , \end{aligned}$$(4.16)provided \(\left( {\textbf{f}},{\textbf{g}}_{T}\right) \in {\textbf{L}}^{2}( \varOmega ) \times {\textbf{L}}_{T}^{2}( \varGamma ) \) for (4.16) and \(\left( {\textbf{f}},{\textbf{g}}_{T}\right) \in {\textbf{X}}_{{\text {imp}}}^{\prime }( \varOmega ) \times {\textbf{X}}_{{\text {imp}}}^{\prime }( \varGamma ) \) for (4.17).$$\begin{aligned} \left\| {\textbf{u}}\right\| _{{\text {imp}},k}&\le C\left( \left\| {\textbf{f}}\right\| _{{\textbf{X}}_{{\text {imp}}}^{\prime }( \varOmega ) ,k}+\left\| {\textbf{g}}_{T}\right\| _{{\textbf{X}}_{{\text {imp}}}^{\prime }( \varGamma ) ,k}\right) , \end{aligned}$$(4.17)
- (iv)Let \(m\in {\mathbb {N}} _{0}\). If \(\varGamma \) is sufficiently smooth and \({\textbf{f}}\in {\textbf{H}} ^{m}({\text {div}},\varOmega )\), \({\textbf{g}}_{T}\in {\textbf{H}}_{T} ^{m+1/2}( \varGamma ) \), then$$\begin{aligned} \left\| {\textbf{u}}\right\| _{{\textbf{H}}^{m+1}\left( \varOmega \right) ,k}&\le C\left| k\right| ^{-3}\left( \left\| {\textbf{f}}\right\| _{{\textbf{H}}^{m}({\text {div}},\varOmega ),k}+\left\| {\textbf{g}} _{T}\right\| _{{\textbf{H}}^{m-1/2}({\text {div}}_{\varGamma },\varGamma ),k}\right) , \end{aligned}$$(4.18a)$$\begin{aligned} \left\| {\textbf{u}}\right\| _{{\textbf{H}}^{m+1}\left( {\text {curl}},\varOmega \right) ,k}&\le C\left| k\right| ^{-2}\left( \left\| {\textbf{f}}\right\| _{{\textbf{H}}^{m} ({\text {div}},\varOmega ),k}+|k|\left\| {\textbf{g}}_{T}\right\| _{{\textbf{H}}^{m+1/2}\left( \varGamma \right) ,k}\right) . \end{aligned}$$(4.18b)
Proof
Proof of (iv): From now on, we assume that \(\varGamma \) is sufficiently smooth. We proceed by induction on \(m\in {\mathbb {N}}_{0}\) and show that if the solution \({\textbf{u}}\in {\textbf{H}}^{m}({\text {curl}},\varOmega )\), then \({\textbf{u}}\in {\textbf{H}}^{m+1}({\text {curl}},\varOmega )\). Specifically, after the preparatory Step 1, we will show \({\textbf{u}}\in {\textbf{H}}^{m+1}(\varOmega )\) in Step 2 and \({\text {curl}}\,{\textbf{u}}\in {\textbf{H}}^{m+1}(\varOmega )\) in Step 3. Step 4 shows the induction hypothesis for \(m=0\) including the norm bounds. Step 5 completes the induction argument for the norm bounds.
5 Regularity Theory for Maxwell’s Equations
In this section, we collect regularity assertions for the Maxwell model problem (2.40). In particular, the case of analytic data studied in Sect. 5.2 will be a building block for the regularity by decomposition studied in Sect. 7.
5.1 Finite Regularity Theory
The difference between Maxwell’s equations with the “good” sign and the time-harmonic Maxwell equations lies in a lower order term. Therefore, higher regularity statements for the solution of Maxwell’s equations can be inferred from those for with the “good” sign, i.e., from Theorem 4.3. The following result makes this precise.
Lemma 5.1
Proof
5.2 Analytic Regularity Theory
Theorem 5.2
Proof
The estimate (5.9) follows from (5.3) of Lemma 5.1 and the definition of the analyticity classes together with the trace estimates \(\Vert {\textbf{g}}_{T}\Vert _{{\textbf{H}}^{1/2}(\varGamma )} \le C C_{{\textbf{g}}} |k|\) and \(\Vert {\textbf{g}}_{T}\Vert _{{\textbf{L}}^{2}(\varGamma )} \le C C_{{\textbf{g}}} |k|^{1/2} \). \(\square \)
6 Frequency Splittings
As in [17, 31, 34, 37–39] we analyze the regularity of Maxwell’s equations (2.40) via a decomposition of the right-hand side into high and low frequency parts.
6.1 Frequency Splittings in \(\varOmega \): \(H_{{\mathbb {R}}^{3}}\), \(L_{{\mathbb {R}}^{3}}\), \(H_{\varOmega }\), \(L_{\varOmega }\), \({H}_{\varOmega }^{0}\), \({L}_{\varOmega }^{0}\)
6.2 Frequency Splittings on \(\varGamma \)
For the definition of the Hodge decompositions and frequency splittings of this section, we recall that \(\varOmega \) has a simply connected, analytic boundary.
Remark 6.1
Remark 6.2
6.3 Estimates for the Frequency Splittings
Lemma 6.3
Proof
The mapping properties for \({\mathcal {L}}_{{\text {imp}}}^{\nabla }\), \({\mathcal {L}}_{{\text {imp}}}^{{\text {curl}}}\), follow directly from elliptic regularity theory on smooth manifolds in view of Remark 6.2. For the stability of the operators \({\textbf{H}}_{\varGamma }^{\nabla }\), \({\textbf{H}}_{\varGamma }^{{\text {curl}}}\) we use the stability of the operator \(H_{\varOmega }:H^{s^{\prime }}(\varOmega )\rightarrow H^{s^{\prime }}(\varOmega )\) for \(s^{\prime }\ge 0\) and the stability of the trace operator \(\gamma :H^{1/2+s^{\prime }}(\varOmega )\rightarrow H^{s^{\prime }}(\varGamma )\) for \(s^{\prime }>0\) as in [38, Lem. 4.2] to get that \({\textbf{h}}_{T}\mapsto \gamma H_{\varOmega }{\mathcal {E}}_{\varOmega }^{\varDelta }{\mathcal {L}}_{{\text {imp}}}^{\nabla }{\textbf{h}}_{T}\) maps continuously \(H^{-1+\varepsilon }(\varGamma )\rightarrow H^{\varepsilon }(\varGamma )\) for any \(\varepsilon >0\) with continuity constant independent of \(\lambda >1\). Since \(\nabla _{\varGamma }:H^{\varepsilon } (\varGamma )\rightarrow {\textbf{H}}_{T}^{-1+\varepsilon }(\varGamma )\), the result follows. The case of \({\textbf{H}}_{\varGamma }^{{\text {curl}}}\) is handled analogously. \(\square \)
We recall some properties of the high frequency splittings that are proved in [38, Lem. 4.2].
Proposition 6.4
- (i)The frequency splitting (6.2) satisfies for all \(0\le s^{\prime }\le s\) the estimates$$\begin{aligned} \left\| H_{{\mathbb {R}}^{3}}f\right\| _{H^{s^{\prime }}({\mathbb {R}}^{3})}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s}\left\| f\right\| _{H^{s}({\mathbb {R}}^{3})}\qquad \forall f\in H^{s}({\mathbb {R}}^{3}), \end{aligned}$$(6.18)$$\begin{aligned} \left\| H_{\varOmega }f\right\| _{H^{s^{\prime }}(\varOmega )}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s}\left\| f\right\| _{H^{s}(\varOmega )}\qquad \forall f\in H^{s} (\varOmega ), \end{aligned}$$(6.19)These estimates hold also for Lipschitz domains.$$\begin{aligned} \left\| H_{\varOmega }^{0}{\textbf{f}}\right\| _{{\textbf{H}}^{s^{\prime } }(\varOmega )}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s}\left\| {\textbf{f}}\right\| _{{\textbf{H}} ^{s}(\varOmega )}\qquad \forall {\textbf{f}}\in {\textbf{H}}^{s}(\varOmega ). \end{aligned}$$(6.20)
- (ii)Let \(0\le s^{\prime }<s\) or \(0<s^{\prime }\le s\). Then the operator \(H_{ \varGamma }\) satisfies$$\begin{aligned} \left\| H_{ \varGamma }g\right\| _{H^{s^{\prime }}( \varGamma )}\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s}\Vert g\Vert _{H^{s}( \varGamma )}. \end{aligned}$$(6.21)
- (iii)Let \(-1\le s^{\prime }<s\) or \(-1<s^{\prime }\le s\). Then the operator \({\textbf{H}}_{\varGamma }\) satisfies for \({\textbf{g}}_{T}\in {\textbf{H}}_{T}^{s}(\varGamma )\)$$\begin{aligned} \left\| {\textbf{H}}_{\varGamma }{\textbf{g}}_{T}\right\| _{{\textbf{H}} ^{s^{\prime }}(\varGamma )}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s}\Vert {\textbf{g}}_{T}\Vert _{{\textbf{H}} ^{s}(\varGamma )}, \end{aligned}$$(6.22a)$$\begin{aligned} \left\| \mathop {{\text {div}}}\nolimits _{\varGamma }\,{\textbf{H}}_{\varGamma } {\textbf{g}}_{T}\right\| _{H^{s^{\prime }-1}(\varGamma )}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s} \Vert \mathop {{\text {div}}}\nolimits _{\varGamma }\,{\textbf{g}}_{T}\Vert _{{\textbf{H}} ^{s-1}(\varGamma )}, \end{aligned}$$(6.22b)$$\begin{aligned} \left\| \mathop {{\text {curl}}}\nolimits _{\varGamma }\,{\textbf{H}}_{\varGamma } {\textbf{g}}_{T}\right\| _{H^{s^{\prime }-1}(\varGamma )}&\le C_{s^{\prime },s}\left( \lambda \left| k\right| \right) ^{s^{\prime }-s} \Vert \mathop {{\text {curl}}}\nolimits _{\varGamma }\,{\textbf{g}}_{T}\Vert _{{\textbf{H}} ^{s-1}(\varGamma )}. \end{aligned}$$(6.22c)
Proof
The following lemma concerns the parameter-explicit bounds for the low frequency operators.
Lemma 6.5
Proof
7 k-Explicit Regularity by Decomposition
In this section, we always assume that the bounded Lipschitz domain \(\varOmega \subset {\mathbb {R}}^{3}\) has a simply connected, analytic boundary \(\varGamma =\partial \varOmega \). We consider the Maxwell problem (2.40) with data \({\textbf{f}}\), \({\textbf{g}}_{T}\) with finite regularity.
- 1.
If \({\text {div}}\,{\textbf{f}} = 0\) (which may be achieved by subtracting a suitable gradient field), then the operator \({\textbf{S}}\) is smoothing by (6.6).
- 2.
The term \({\mathcal {S}}_{\varOmega ,k}^{+}(H_{\varOmega }^{0} {\textbf{f}} + H_{\varOmega }{\textbf{S}}{\textbf{f}},{\textbf{H}}_{\varGamma }{\textbf{g}}_{T})\) has finite regularity properties given by Theorem 4.3. The effect of the high frequency filters \(H_{\varOmega }^{0}\) and \({\textbf{H}}_{\varGamma }\) is that they improve the k-dependence of lower-order terms in the indexed norms such as \(\Vert \cdot \Vert _{{\textbf{H}}^{m}(\varOmega ),k}\) (see Lemma 7.1 below).
- 3.
\({\mathcal {S}}_{\varOmega ,k}^{{\text {MW}}}(L_{\varOmega }^{0} {\textbf{f}} + L_{\varOmega }{\textbf{S}} {\textbf{f}},{\textbf{L}}_{\varGamma }{\textbf{g}}_{T})\) is an analytic function and can be estimated with the aid of Theorem 5.2.
- 4.The function \({\textbf{z}}^{\prime }\) satisfieswhere, by suitably choosing the cut-off parameters \(\lambda \) in the frequency operators, the residual \({\textbf{r}}\) satisfies \(\Vert {\textbf{r}}\Vert _{*}\le q\Vert {\textbf{f}}\Vert _{*}\) for some \(q\in (0,1)\) and a suitable norm \(\Vert \cdot \Vert _{*}\). Hence, the arguments can be repeated for \({\textbf{z}}^{\prime }\) and the decomposition can be obtained by a geometric series argument.$$\begin{aligned} {\mathcal {L}}_{\varOmega ,k}^{{\text {MW}}}{\textbf{z}}^{\prime }={\textbf{r}},\qquad {\mathcal {B}}_{ \varGamma ,k}=0, \end{aligned}$$
7.1 The Concatenation of \({\mathcal {S}}_{\varOmega ,k}^{+}\) with High Frequency Filters
The following lemma analyzes the mapping properties of the concatenation of the solution operator \({\mathcal {S}}_{\varOmega ,k}^{+}\) with the high frequency filter operators \(H_{\varOmega }^{0}\) and \({\textbf{H}}_{\varGamma }\).
Lemma 7.1
Proof
Proof of (7.1): (7.1) follows from the fact that \({\text {div}} H_{\varOmega }^{0} {\textbf{f}} = 0\) and Proposition 6.4.
Proof of (7.4): For \(m\ge 1\), we obtain (7.4) from (7.2) and Theorem 4.3, (4.18a). For \(m=0\), we observe that Theorem 4.3, (4.17) and (7.3) imply \(\Vert {\mathcal {S}}_{\varOmega ,k} ^{+}(H_{\varOmega }^{0}{\textbf{f}},0)\Vert _{{\textbf{L}}^{2}(\varOmega )}\le |k|^{-1}\Vert {\mathcal {S}}_{\varOmega ,k}^{+}(H_{\varOmega }^{0}{\textbf{f}},0)\Vert _{{\text {imp}},k}\le C{|k|^{-1}}\Vert H_{\varOmega }^{0} {\textbf{f}}\Vert _{{\textbf{X}}_{{\text {imp}}}^{\prime }( \varOmega ),k} \le C\lambda ^{-1/2}|k|^{-2}\Vert {\textbf{f}}\Vert _{{\textbf{L}}^{2}(\varOmega )}\).
Proof of (7.5): We distinguish the cases \(m=0\) and \(m\ge 1\). For \(m\ge 1\), the statement follows from the estimates of \({{\textbf{H}}}_{\varGamma }\) given in Proposition 6.4. For \(m=0\), in addition to Proposition 6.4 one invokes Lemma 4.2.
Proof of (7.7), (7.8): For \({\textbf{f}}\) with \({\text {div}}\,{\textbf{f}}=0\) we have \({\textbf{S}}{\textbf{f}}\in C^{\infty }( \varOmega ) \) by (6.6). Hence, (7.7), (7.8) follow from Proposition 6.4 and (6.6) and again Theorem 4.3. \(\square \)
Lemma 7.2
Proof
7.2 Regularity by Decomposition: The Main Result
Theorem 7.3
Let \(\varOmega \subset {\mathbb {R}}^{3}\) be a bounded Lipschitz domain with a simply connected, analytic boundary \(\varGamma =\partial \varOmega \). Let the stability Assumption (3.2) be satisfied. Then there is a linear mapping \({\textbf{L}}^{2}(\varOmega )\times {\textbf{H}}_{T}^{-1/2} ({\text {div}}_{\varGamma },\varGamma )\ni ({\textbf{f}},{\textbf{g}}_{T} )\mapsto ({\textbf{z}}_{H^{2}},{\textbf{z}}_{{\mathcal {A}}},\varphi _{{\textbf{f}} },\varphi _{{\textbf{g}}})\) such that the solution \({\textbf{z}}:={\mathcal {S}} _{\varOmega ,k}^{{\text {MW}}}({\textbf{f}},{\textbf{g}}_{T})\in {\textbf{X}} _{{\text {imp}}}\) of (2.40) can be written as \({\textbf{z}} ={\textbf{z}}_{H^{2}}+{\textbf{z}}_{{\mathcal {A}}}+k^{-2}\nabla \varphi _{{\textbf{f}} }+{\text {i}}\,k^{-1}\nabla \varphi _{{\textbf{g}}}\).
- (i)The function \({\textbf{z}}_{H^{2}}\) satisfiesIf \({\textbf{g}}_{T}\in {\textbf{H}}_{T}^{m+1/2}(\varGamma )\), then$$\begin{aligned} \Vert {\textbf{z}}_{H^{2}}\Vert _{{\textbf{H}}^{m+1}(\varOmega ),k}\le C|k|^{-m-2} \left( |k|\Vert {\textbf{g}}_{T}\Vert _{{\textbf{H}}^{m-1/2}(\varGamma )} +\Vert {\textbf{f}}\Vert _{{\textbf{H}}^{m}(\varOmega )}\right) . \end{aligned}$$(7.10)and in (7.10) the term \(|k|\Vert {\textbf{g}}_{T}\Vert _{{\textbf{H}}^{m-1/2}(\varGamma )}\) can be replaced with \(\Vert {\textbf{g}}_{T} \Vert _{{\textbf{H}}^{m+1/2}(\varGamma )}\).$$\begin{aligned} \left\| {\textbf{z}}_{H^{2}}\right\| _{{\textbf{H}}^{m+1}\left( {\text {curl}},\varOmega \right) ,k}\le C\left| k\right| ^{-m-1}\left( \left\| {\textbf{g}}_{T}\right\| _{{\textbf{H}} ^{m+1/2}\left( \varGamma \right) }+\left\| {\textbf{f}}\right\| _{{\textbf{H}}^{m}\left( \varOmega \right) }\right) \end{aligned}$$(7.11)
- (ii)The gradient fields \(\nabla \varphi _{{\textbf{f}}}\) and \(\nabla \varphi _{{\textbf{g}}}\) are given by Lemma 7.2 and satisfy, for \(\ell =0,\ldots ,m^{\prime }\):$$\begin{aligned} \left\| \varphi _{{\textbf{f}}}\right\| _{{\textbf{H}}^{\ell +1}(\varOmega )}&\le C\left\| {\text {div}}\,{\textbf{f}}\right\| _{H^{\ell -1}\left( \varOmega \right) }, \end{aligned}$$(7.12)$$\begin{aligned} \left\| \varphi _{{\textbf{g}}}\right\| _{{\textbf{H}}^{\ell +1}(\varOmega )}&\le C\left\| {\text {div}}_{\varGamma }\,{\textbf{g}}_{T}\right\| _{H^{\ell -3/2}\left( \varGamma \right) }. \end{aligned}$$(7.13)
- (iii)The analytic part \({\textbf{z}}_{{\mathcal {A}}}\) satisfies$$\begin{aligned} {\textbf{z}}_{{\mathcal {A}}}\in {\mathcal {A}}(C{(1+C_\mathrm{{stab}})}|k|^{ \theta -1} \{\Vert {\textbf{f}}\Vert +|k|\Vert {\textbf{g}}_{T}\Vert _{{\textbf{H}}^{-1/2}(\varGamma )}\},B,\varOmega ). \end{aligned}$$(7.14)
Proof
By linearity of the solution operator \({\mathcal {S}}_{\varOmega ,k} ^{{\text {MW}}}\), we consider the cases \({\mathcal {S}}_{\varOmega ,k}^{{\text {MW}}}({\textbf{f}},0)\) and \({\mathcal {S}}_{\varOmega ,k} ^{{\text {MW}}}(0,{\textbf{g}}_{T})\) separately. The fact that the right-hand sides in (7.10), (7.11), (7.14) do not contain the divergence of \({\textbf{f}}\) or \({\textbf{g}}_{T}\) is due to the fact that we suitably choose the functions \(\varphi _{{\textbf{f}}}\), \(\varphi _{{\textbf{g}}}\) in the course of the proof.
8 Discretization
In this section, we describe the hp-FEM based on Nédélec elements and discuss the approximation properties of various hp-approximation operators. These operators made their appearance already in [39]. Here, we strengthen the results of [39, Sec. 8] in that we additionally control the error on the boundary of the elements, which is required due to the impedance boundary conditions considered here.
8.1 Meshes and Nédélec Elements
- (i)
The (open) elements \(K\in {{\mathcal {T}}}_{h}\) cover \(\varOmega \), i.e., \(\overline{\varOmega }=\cup _{K\in {{\mathcal {T}}}_{h}}{\overline{K}}\).
- (ii)
Associated with each element K is the element map, a \(C^{1} \)-diffeomorphism \(F_{K}:\overline{{\widehat{K}}} \rightarrow {\overline{K}}\). The set \({\widehat{K}}\) is the reference tetrahedron.
- (iii)Denoting \(h_{K}={\text {diam}}K\), there holds, with some shape-regularity constant \(\gamma _{{\mathcal {T}}}\),$$\begin{aligned} h_{K}^{-1}\Vert F_{K}^{\prime }\Vert _{L^{\infty }({\widehat{K}})}+h_{K}\Vert (F_{K}^{\prime })^{-1}\Vert _{L^{\infty }({\widehat{K}})}\le \gamma _{{\mathcal {T}}}. \end{aligned}$$(8.1)
- (iv)
The intersection of two elements is only empty, a vertex, an edge, a face, or they coincide (here, vertices, edges, and faces are the images of the corresponding entities on the reference tetrahedron \({\widehat{K}}\)). The parametrization of common edges or faces are compatible. That is, if two elements K, \(K^{\prime }\) share an edge (i.e., \(F_{K}(e)=F_{K^{\prime } }(e^{\prime })\) for edges e, \(e^{\prime }\) of \({\widehat{K}}\)) or a face (i.e., \(F_{K}(f)=F_{K^{\prime }}(f^{\prime })\) for faces f, \(f^{\prime }\) of \({\widehat{K}}\)), then \(F_{K}^{-1}\circ F_{K^{\prime }}:f^{\prime }\rightarrow f\) is an affine isomorphism.
Assumption 8.1
Remark 8.2
A prime example of meshes that satisfy Assumption 8.1 are those patchwise structured meshes as described, for example, in [37, Ex. 5.1] or [33, Sec. 3.3.2]. These meshes are obtained by first fixing a macro triangulation of \(\varOmega \); the actual triangulation is then obtained as images of affine triangulations of the reference element. \(\square \)
8.2 hp-Approximation Operators
We will use polynomial approximation operators that are constructed elementwise, i.e., for an operator \({\widehat{I}}_{p}\) on the reference element \({\widehat{K}}\), a global operator \(I_{p}\) is defined by setting \((I_{p}u)|_{K}:={\widehat{I}}_{p}(u\circ F_{K}))\circ F_{K}^{-1} \). If \({\widehat{I}}_{p}\) maps into \({{\mathcal {P}}}_{p+1}({\widehat{K}})\), then we say \({\widehat{I}}_{p}\) admits an element-by-element construction, if the operator \(I_{p}\) defined in this way maps into \(S_{p+1}({{\mathcal {T}}}_{h})\). Analogously, if \({\widehat{I}}_{p}\) maps into \(\varvec{{\mathcal {N}}} _{p}^{{\text {I}}}({\widehat{K}})\), then we say that \({\widehat{I}}_{p}\) admits an element-by-element construction if the resulting operator \(I_{p}\) maps into \(\varvec{{\mathcal {N}}}_{p}^{{\text {I}} }({\mathcal {T}}_{h})\).
For scalar functions (or gradient fields), we have elemental approximation operators with the optimal convergence in \(L^{2}\) and \(H^{1}\):
Lemma 8.3
For the case \(d=3\), the condition on m can be relaxed to \(m >d/2\).
Proof
The operator \({\widehat{\varPi }}_{p}\) may be taken as the operators \(\widehat{\varPi }^{{\text {grad}},3d}_{p+1}\) for \(d = 3\) or \(\widehat{\varPi }^{{\text {grad}},2d}_{p+1}\) for \(d = 2\) of [36]. The volume estimates follow from [36, Cor. 2.12] for the case \(d= 3\) and [36, Thm. 2.13] for the case \(d = 2\). For the estimates on \(\partial {\widehat{K}}\), one notices that the restriction of \({\widehat{\varPi }}^{{\text {grad}},3d}_{p+1}\) to a boundary face \({\widehat{f}}\) is the operator \({\widehat{\varPi }}^{{\text {grad}},2d}_{p+1}\) on that face and that the restriction of \({\widehat{\varPi }}^{{\text {grad}},2d}_{p+1}\) to an edge of the reference triangle is the operator \(\widehat{\varPi }^{{\text {grad}},1d}_{p+1}\) discussed in [36, Lem. 4.1].
For \(d=3\) an operator \({\widehat{\varPi }}_{p}\) with the stated approximation properties is constructed in [37, Thm. B.4] for the case \(m >d/2=3/2\). The statement about the approximation on \(\partial {\widehat{K}}\) follows by a more careful analysis of the proof of [37, Thm. B.4]. For the reader’s convenience, the proof is reproduced in [40, Thm. B.5]. \(\square \)
The fact that \({\widehat{\varPi }}_{p}\) in Lemma 8.3 has the element-by-element construction property means that an elementwise definition of the operator \(\varPi _{p}^{\nabla ,s}:H^{m}(\varOmega )\rightarrow S_{p+1}({\mathcal {T}}_{h})\) by \((\varPi _{p}^{\nabla ,s}\varphi )|_{K}=({\widehat{\varPi }}_{p}(\varphi \circ F_{K}))\circ F_{K}^{-1}\) maps indeed into \(S_{p+1}({\mathcal {T}}_{h})\subset H^{1}(\varOmega )\).
In the following we always assume for the spatial dimension \(d=3\). By scaling arguments we get the following result:
Corollary 8.4
Lemma 8.5
- (i)If \({\textbf{u}}\in {\textbf{H}}^{m}(K)\) then$$\begin{aligned}&\Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}},s}{\textbf{u}}\Vert _{{\textbf{L}} ^{2}(K)}+\frac{h_{K}}{p+1}\Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}} ,s}{\textbf{u}}\Vert _{{\textbf{H}}^{1}(K)} \le C\left( \frac{h_{K}}{p+1}\right) ^{m} \Vert {\textbf{u}}\Vert _{{\textbf{H}}^{m}(K)}, \end{aligned}$$(8.15)$$\begin{aligned}&\Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}},s}{\textbf{u}}\Vert _{{\textbf{L}} ^{2}(\partial K)} \le C\left( \frac{h_{K}}{p+1}\right) ^{m-1/2} \Vert {\textbf{u}}\Vert _{{\textbf{H}}^{m}(K)}. \end{aligned}$$(8.16)
- (ii)If \({\textbf{u}}\in {\mathcal {A}} (C_{{\textbf{u}}}(K),B,K)\) for some \(C_{{\textbf{u}}}(K)>0\) and ifthen$$\begin{aligned} h_{K}+|k|h_{K}/p\le {\widetilde{C}} \end{aligned}$$(8.17)$$\begin{aligned}&h_{K}^{1/2}\Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}},s}{\textbf{u}} \Vert _{{\textbf{L}}^{2}(\partial {K})}+\Vert {\textbf{u}}-\varPi _{p} ^{{\text {curl}},s}{\textbf{u}}\Vert _{{\textbf{L}}^{2}({K})}+h_{K} \Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}},s}{\textbf{u}}\Vert _{{\textbf{H}} ^{1}({K})}\nonumber \\&\quad \le CC_{{\textbf{u}}}(K)\left( \left( \frac{h_{K}}{h_{K}+\sigma }\right) ^{p+1}+\left( \frac{|k|h_{K}}{\sigma p}\right) ^{p+1}\right) . \end{aligned}$$(8.18)
- (iii)If \({\textbf{u}}\in {\mathcal {A}} (C_{{\textbf{u}}},B,\varOmega )\) for some \(C_{{\textbf{u}}}>0\) and if (8.17) holds, then$$\begin{aligned} \Vert {\textbf{u}}-\varPi _{p}^{{\text {curl}},s}{\textbf{u}}\Vert _{{\text {imp}},k}\le C_{{\textbf{u}}}|k|\left( \left( \frac{h}{h+\sigma }\right) ^{p}+\left( \frac{|k|h}{\sigma p}\right) ^{p}\right) . \end{aligned}$$
Proof
8.3 An Interpolating Projector onto the Finite Element Space
Proposition 8.6
The proof of this proposition is standard and uses the same arguments as, e.g., [39, Lem. 4.10].
Proposition 8.7
Proof
Lemma 8.8
- (i)
\(\varPi _{h}^{E}\) is a projection, i.e., the restriction \(\left. \varPi _{h}^{E}\right| _{{\textbf{X}}_{h}}\) is the identity on \({\textbf{X}}_{h}\).
- (ii)
The operators \(\varPi _{h}^{E}\) and \(\varPi _{h}^{F}\) have the commuting property: \({\text {curl}}\,\varPi _{h}^{E}=\varPi _{h}^{F}{\text {curl}}\).
Proof
Since \(\varPi _{p}^{{\text {curl}},c}\) is based on an element-by-element construction it is well defined on \({\textbf{H}}({\text {curl}},\varOmega )\cap {\prod _{K\in {\mathcal {T}}_{h}}}{\textbf{H}}^{1}({\text {curl}},K)\). Since \({\textbf{V}}_{k,0,h}+{\textbf{X}}_{h}\) is a subspace of this space, the mapping properties follow. The projection property of \(\varPi _{h}^{E}\) and the commuting property of \(\varPi _{h}^{E}\) and \(\varPi _{h}^{F}\) are proved in [36, Thm. 2.10, Rem. 2.11]. \(\square \)
9 Stability and Convergence of the Galerkin Discretization
The wavenumber-explicit stability and convergence analysis for Maxwell’s equations with transparent boundary conditions has been developed recently in [39] and generalizes the theory in [41, Sec. 7.2]. A “roadmap” for the convergence proof of [39] is given in [39, Sec. 1.1–1.3]. In the present analysis, we follow this “roadmap” taking into account the change in boundary conditions from transparent boundary conditions to impedance boundary conditions. A key role is played by the term \({ \left( \!\left( \hspace{-0.50003pt} {\textbf{u}},{\textbf{v}} \hspace{-1.00006pt}\right) \!\right) }_k = A_{k}( {\textbf{u}},{\textbf{v}}) - \left( {\text {curl}} \textbf{u,}\,{\text {curl}}\,{\textbf{v}}\right) \) from (2.42), which includes the boundary conditions. This sesquilinear form determines the space \({\textbf{V}}_{k,0}\) (see (8.22)) and the regular decomposition in Def. 9.2 ahead and its properties differentiate the present case of impedance boundary conditions from the transparent boundary condition case. Compared to the case of transparent boundary conditions, the present impedance boundary conditions case is simpler in that fewer approximation quantities \(\eta ^{{\text {alg}}}_j\), \({{\tilde{\eta }}}_j^{{\text {alg}}}\) are required in the analysis.
In this section, we develop a stability and convergence theory for Maxwell’s equations with impedance boundary conditions, see Sect. 2.5. Recall the definition of the sesquilinear form \( { \left( \!\left( \hspace{-0.50003pt} \cdot ,\cdot \hspace{-1.00006pt}\right) \!\right) }_k\) of (2.42) and of the norm \(\left\| \cdot \right\| _{k,+}\) in Definition 2.5.
Proposition 9.1
Proof
9.1 Splitting of the Consistency Term
We introduce continuous and discrete Helmholtz decompositions that are adapted to the problem under consideration.
Definition 9.2
Solvability of these equations follows trivially from the Lax-Milgram lemma as can be seen from the following lemma.
Lemma 9.3
Proof
9.2 Consistency Analysis: The Term \({\textbf{T}}_{1}\) in (9.9)
9.2.1 hp-Analysis of \(T_{1}\)
Lemma 9.4
Proof
9.3 Consistency Analysis: The Term \({\textbf{T}}_{2}\) in (9.9)
9.3.1 hp-Analysis of \(T_{2}\)
Lemma 9.5
Proof
9.4 h-p-k-Explicit Stability and Convergence Estimates for Maxwell’s Equations
We begin with the estimate of the consistency term \(\delta _{k}\).
Lemma 9.6
Proof
This estimate allows us to formulate the quasi-optimality of the hp-FEM Galerkin discretization and to show h-p-k-explicit convergence rates under suitable regularity assumptions.
Theorem 9.7
Let \(\varOmega \subset {\mathbb {R}}^{3}\) be a bounded Lipschitz domain with a simply connected, analytic boundary. Let the stability Assumption (3.2) be satisfied. Let the finite element mesh with mesh size h satisfy Assumption 8.1, and let \({\textbf{X}}_{h}\) be defined by as the space of Nédélec-type-I elements of degree p (cf. (8.7)).
Then, for any \({\textbf{j}}\), \({\textbf{g}}_{T}\) satisfying (2.38), the variational form of Maxwell’s equations (2.39) has a unique solution \({\textbf{E}}\).
Proof
Existence and uniqueness of the continuous variational Maxwell problem follow from Proposition 3.1. From Lemma 9.6 we know that \(c_{1}\) can be chosen sufficiently small such that \(\delta ({\textbf{e}}_{h})<\eta \). As in the proof of Theorem [39, Thm. 4.15] (which goes back to [27, Thm. 3.9]) existence, uniqueness, and quasi-optimality follows. \(\square \)
The quasi-optimality result (9.36) leads to quantitative, k-explicit error estimates if a k-explicit regularity of the solution \({\textbf{E}}\) is available. In the following corollary, we draw on the regularity assertions of Theorem 7.3. We point out, however, that due to our relying on the operator \(\varPi _{p}^{{\text {curl}},s}\) and the regularity assertion Theorem 7.3, the regularity requirements on the data \({\textbf{j}}\), \({\textbf{g}}_{T}\) are not the weakest possible ones.
Corollary 9.8
Proof

\(\varOmega =(-1,1)^{3}\), smooth solution; left to right: \(p\in \{1,2,3\}\)

\(\varOmega = (-1,1)^{3}\setminus [-1/2,1/2]^{3}\), smooth solution; left to right: \(p \in \{1,2,3\}\)
10 Numerical Results
We illustrate the theoretical findings of Theorem 9.7 and Corollary 9.8 by two numerical experiments. All computations are performed with NGSolve, [47, 48] using Nédélec type II elements, i.e., full polynomial spaces.
Remark 10.1
While the analysis of the present paper is peformed in detail for Nédélec type I elements, it can be extended to Nédélec type II elements. Key is the observation that commuting diagram operators \({\widehat{\varPi }}^{grad,c}_{p+1}\) and \({\widehat{\varPi }}^{curl,c}_p\) analogous to the ones used in Sects. 8 and 9 for type I elements also exist for type II elements. This is discussed in [46, Sec. 4.8].\(\square \)
Example 10.2
We consider \(\varOmega =(-1,1)^{3}\) and impose the right-hand side and the impedance boundary conditions in such a way that the exact solution is \({\textbf{E}}\left( {\textbf{x}}\right) ={\text {curl}}\sin (kx_{1} )(1,1,1)^{\top }\). Figure 1 shows the performance for the choices \(k\in \{10,20,30,40\}\) and \(p\in \{1,2,3\}\) as the mesh is refined quasi-uniformly. The final problem sizes were \({\text {DOF}}=18,609,324\) for \(p=1\), \({\text {DOF}}=9,017,452\) for \(p=2\), and \({\text {DOF}}=23,052,940\) for \(p=3\).
We observe the expected asymptotic \(O(h^{p})\) convergence. We also observe that the onset of asymptotic quasi-optimal convergence is reached for smaller values of \(N_{\lambda }\) for higher order methods. This is expected in view of Theorem 9.7, although the present setting of a piecewise analytic geometry is not covered by Theorem 9.7. \(\square \)
Example 10.3
We consider \(\varOmega =(-1,1)^{3}{\setminus } [-1/2,1/2]^{3}\) and Maxwell’s equations with impedance boundary conditions on \(\partial (-1,1)^{3}\) and perfectly conducting boundary conditions on the inner boundary \(\partial (-1/2,1/2)^{3}\). We prescribe an exact solution \({\textbf{E}}( {\textbf{x}}) =k\cos (kx_{1})(x_{1}^{2}-1/4)(x_{2} ^{2}-1/4)(x_{3}^{2}-1/4)(0,-1,1)^{\top }\). Figure 2 shows the performance for the choices \(k\in \{20,40,80\}\) and \(p\in \{1,2,3\}\) as the mesh is refined quasi-uniformly. The final problem sizes were \({\text {DOF}}=43,598,374\) for \(p=1\), \({\text {DOF}}=168,035,046\) for \(p=2\), and \({\text {DOF}} =54,063,558\) for \(p=3\).
We observe the expected asymptotic \(O(h^{p})\) convergence. We also observe that the onset of asymptotic quasi-optimal convergence is reached for smaller values of \(N_{\lambda }\) for higher order methods. \(\square \)
Acknowledgements
We cordially thank Claudio Rojik (TU Wien) for assistance with the numerical computations in Sect. 10. Financial support by the Austrian Science Fund FWF (through Grants P 28367-N35 and F65) is gratefully acknowledged.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.