Abstract
We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.
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Research of A. Cesmelioglu was supported by the Oakland University URC Faculty Research Fellowship Award. Research of J.J.W. Van der Vegt was supported by the High-end Foreign Experts Recruitment Program (GDW20157100301), while the author was in residence at the University of Science and Technology of China in Hefei, Anhui, China. Research of Yan Xu was supported by NSFC Grant Nos. 11371342 and 11526212.
Appendices
Appendix
Appendix A: Continuity and Semi-Ellipticity
Lemma 10.1
For all \(\varvec{v}\in {\varvec{V}}(h)\) and \(q \in Q(h)\),
with a constant \(C>0\) that is independent of the mesh size and the coefficient \(\epsilon \).
Proof
Here we use the inverse inequality Lemma 11 in [41]:
The proof of the other estimate is similar. \(\square \)
Theorem 10.1
There exist constants \(a_1>0\) and \(a_2>0\), independent of the mesh size and the coefficient \(\epsilon \), such that
Proof
\(\square \)
Lemma 10.2
For \({\varvec{\alpha }}\in K \text { with } {\varvec{\alpha }}\not = \varvec{{0}}\), given that \({\mathfrak {a}}>0\) is large enough, \({\mathfrak {b}}>0\) and \({\mathfrak {c}}>0\), there exists a \(C>0\) independent of h, such that
Proof
where we used the estimate in Lemma 10.1. \(\square \)
Appendix B: Inf-Sup Condition
For the proof of Lemma 5.5, we first need the following auxiliary result.
Lemma 10.3
Given N real numbers \(\{\alpha _1,\dots ,\alpha _N\}\) let \( \beta =\frac{1}{N}\sum _{j=1}^{N}\alpha _j\). Then,
where \( C>0 \) depends only on N.
Proof
For any j, the Cauchy-Schwarz inequality gives
Summing over j, we obtain
\(\square \)
Proof of Lemma 5.5
Given \( q_h\in Q^{\varvec{\alpha }}_h\), we construct a function \( \chi \in Q^{{\varvec{\alpha }},c}_h\) as follows: At every node of the mesh \( {\mathcal {T}_h }\) corresponding to a Lagrangian type degree of freedom for \( Q^{{\varvec{\alpha }},c}_h\), the value of \( \chi \) is set to the average of the values of \( q_h\) at that node.
For each \(K\in {\mathcal {T}_h }\), let \(\mathcal {N}_K=\{ \varvec{x}^{(j)}_K,j=1,\ldots ,m \}\) be the Lagrange nodes (points) of K and \( \{ \phi ^{(j)}_K,j=1,\ldots ,m \} \) the corresponding (local) basis functions satisfying \( \phi ^{(j)}_K(\varvec{x}^{(i)}_K)=\delta _{ij} \). Set \( \mathcal {N}=\cup _{K\in {\mathcal {T}_h }} \mathcal {N}_K\). We view \( \mathcal {N}\) as the union of two classes:
We note that \(\mathcal {N}_f\) can be divided into two sets \(\mathcal {N}_f^i\) and \(\mathcal {N}_f^b\): \(\mathcal {N}_f^i\) is the set of nodes on interior faces, while \(\mathcal {N}_f^b\) is the set of nodes on the boundary of a face \(f \subset \partial \varOmega \). As \(\varOmega \) and \({\mathcal {T}_h }\) are both periodic, for every \(\nu ^1\in \mathcal {N}_f^b\), there exist a unique \(\nu ^2\) also in \(\mathcal {N}_f^b\) being the corresponding periodic point of \(\nu ^1\). From the definition \(Q^{{\varvec{\alpha }},c}_h=Q^{\varvec{\alpha }}_h\cap {H^1_{\mathrm {per}}(\varOmega ) }\), \(Q^{{\varvec{\alpha }},c}_h\) is a periodic conforming finite element space. To satisfy the periodicity of \(Q^{{\varvec{\alpha }},c}_h\), we can let \(\nu ^1\) and \(\nu ^2\) share the same degree of freedom. Then we regard \(\nu ^1\) and \(\nu ^2\) as the ’same’ node in our computational domain. Furthermore, the nodes in \(\mathcal {N}_f^b\) can therefore be considered as nodes in \(\mathcal {N}_f^i\). In the following discussion, we consider the nodes in \(\mathcal {N}_f^b\) and \(\mathcal {N}_f^i\) therefore in the same way.
For each \( \nu \in \mathcal {N}\), let \( \omega _\nu =\{K \in {\mathcal {T}_h }| \nu \in K \} \) and denote its cardinality by \( \left| \omega _\nu \right| \). If \(\nu \in \mathcal {N}_i\), then \( \left| \omega _\nu \right| =1 \), and if \(\nu \in \mathcal {N}_f\), \(\left| \omega _\nu \right| \ge 1\). Then the basis function \( \phi ^{(\nu )}\) in \(Q^{{\varvec{\alpha }},c}_h\) at the node \(\nu \in \mathcal {N}\) can be constructed as
Now, given \( q_h\in Q^{\varvec{\alpha }}_h\), written as \( q_h=\sum _{K\in {\mathcal {T}_h }} \sum _{j=1}^{m}\alpha ^{(j)}_K\phi ^{(j)}_K\), we define the function \( \chi \in Q^{{\varvec{\alpha }},c}_h\) by
where
Now set \( \beta ^{(j)}_K=\beta ^{(\nu )}\) whenever \( \varvec{x}^{(j)}_K=\nu \). A simple scaling argument shows that \( \Vert \nabla _{\varvec{\alpha }}\phi ^{(j)}_K \Vert ^2_K \le ch^{d-2}_K\). Hence
where in the last step, we remove the nodes in \(\mathcal {N}_i\) as they have no contribution by the definition of \(\beta ^{(\nu )}\).
We now temporarily focus on the case \( d = 2 \). For \( \nu \in \mathcal {N}_f\) we enumerate the elements of \( \omega _\nu \) as \( \{ K_{1}, \dots , K_{\left| \omega _\nu \right| } \} \) so that any consecutive pair \( K_{i} \), \(K_{i+1}\) in that list shares an edge. Then from Lemma 10.3, with some constant C depending only on \( \left| \omega _\nu \right| \), we have
For \( d=3 \), it may not be possible to enumerate \( \omega _\nu \) in such a way. However, by allowing some repetitions of its elements, we can write \( \omega _\nu =\{ K_{l_1}, \ldots ,K_{l_{n(\nu )}} \} \) for some \( n(\nu )\), so that in this case also \( K_{l_i} \) and \( K_{l_{i+1}} \) share a face or an edge. Having done so, by applying Lemma 10.3 to the list obtained by removing all repetitions of elements of \( \omega _\nu \), we obtain
Using (10.4) for \(d=2\), or (10.5) if \(d=3\), from (10.3) we have
with \( \varvec{x}^{j^{+}_{\nu }}_{K_+}=\varvec{x}^{j^{-}_{\nu }}_{K_-}=\nu \). Note that \( \alpha ^{j^{+}_{\nu }}_{K_+}-\alpha ^{j^{-}_{\nu }}_{K_-} \) is the jump in the values of \(q_h\) at \(\nu \) across f. Also, since the mesh \({\mathcal {T}_h }\) is locally quasi-uniform, it follows that
where the constant C depends on the number of nodes in f . The required result now follows from (10.6)–(10.7). \(\square \)
Proof of Theorem 5.2
Fix \(0 \not = q \in Q^{\varvec{\alpha }}_h\), and use the \(Q^{\varvec{\alpha }}_h\)-decomposition as \(q = q_0 + q_1\) with \(q_0\in Q^{{\varvec{\alpha }},c}_h\) and \(q_1\in Q_h^{{\varvec{\alpha }},\perp }\). Choose \(\varvec{v}_0= - \nabla _{\varvec{\alpha }}q_0 \in {\varvec{V}}^{\varvec{\alpha }}_h\cap {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl}^0_{\varvec{\alpha }};\varOmega ) }\), then we have
where we use \( \Vert \nabla _{\varvec{\alpha }}\phi \Vert _{0, \partial K}\le C h_K^{- \frac{1}{2}}\Vert \nabla _{\varvec{\alpha }}\phi \Vert _{0,K}\), for any \(\phi = {e^{i{\varvec{\alpha }}\cdot \varvec{x}}}\tilde{\phi }\) and \( \tilde{\phi }\in \mathcal {S}_l(K)\) with \(C>0\), which we obtain from the trace inequality \( \Vert \nabla \tilde{\phi }\Vert _{0, \partial K}\le C h_K^{- \frac{1}{2}}\Vert \nabla \tilde{\phi }\Vert _{0,K}\), with \(C>0\). Let \({\varvec{\nu }}_1= - {[[ q_1 ]] _N}\). Using Lemma 5.6, we obtain
Let \((\varvec{v}, {\varvec{\nu }})=(\varvec{v}_0,\varvec{{0}})+\delta (\varvec{{0}}, {\varvec{\nu }}_1)\) with \(\delta >0\). Since \(q_0\in Q^{{\varvec{\alpha }},c}_h\), \({[[ q_0 ]] _N} =\varvec{{0}}\) on \({\mathcal {F}_h }\) and \(B_h(\varvec{{0}},{\varvec{\nu }}_1;q_0)=c \int _ {{\mathcal {F}_h }}{{h}}^{-1}{[[ q_0 ]] _N}\cdot \bar{{\varvec{\nu }}}_1 ds =0\), we have
Using Theorem 10.1 and (10.9), we obtain
with any \(\zeta >0\). Choosing suitable \(\delta \) and \(\zeta \), we have
with \(k_1>0\). From (10.9) and (10.10), we have
Then the result follows with \(k=k_1/k_2\). \(\square \)
Appendix C: Ellipticity on the Kernel
Lemma 10.4
Proof
Let \(\varvec{v}\in {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\). By [16, Theorem 3.1], there exists \(\varvec{w}\in {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\) and \( \phi \in {H^1_{\mathrm {per}}(\varOmega ) }\) such that
By Lemma 3.1, since \(\nabla _{\varvec{\alpha }}\cdot \nabla _{\varvec{\alpha }}\times \varvec{v}=0\), we obtain \(\phi =0\). Therefore
implying \(\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\subset \nabla _{\varvec{\alpha }}\times {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\). The other inclusion is obvious as \({{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\subset {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\). \(\square \)
Proof of Lemma 5.7
Lemma 10.4 implies that \(\nabla _{\varvec{\alpha }}\times \) maps \({{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\) onto \(\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\). Let K denote the orthogonal complement of the kernel of \(\nabla _{\varvec{\alpha }}\times \) in \({{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\). Then, the restriction \(\nabla _{\varvec{\alpha }}\times |_{K}\) of \(\nabla _{\varvec{\alpha }}\times \) to K also maps \({{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\) onto \(\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\). In addition to being onto, \(\nabla _{\varvec{\alpha }}\times |_{K}\) is continuous, one-to-one and has a continuous inverse due to [16, Theorem 3.1]. The operator \(R=(\nabla _{\varvec{\alpha }}\times |_{K})^{-1}\nabla _{\varvec{\alpha }}\times \) satisfies the conclusion of the lemma. \(\square \)
Lemma 10.5
For \(\varvec{u}\in {\varvec{L}}^2(\varOmega )\), we have the following estimate for the auxiliary problem (10.12):
Proof
Taking the periodic boundary conditions into consideration and integrating by parts, we have
where \(\nabla _{\varvec{\alpha }}\times (\nabla _{\varvec{\alpha }}\psi )=\varvec{{0}}\). Combining with the estimate given in Theorem 3.3 gives the result. \(\square \)
Proof of Theorem 5.3
From the seminorm ellipticity in Lemma 10.2 , it is sufficient to show that there exist \(C>0\), such that
Now fix \((\varvec{u}, {\varvec{\nu }}) \in {\mathrm {Ker}(B_h)}\), and let \((\varvec{z}, \psi )\in {\varvec{V}}\times Q \) satisfying
with periodic boundary conditions. Thereby,
where the detailed derivation of (10.13) is given in Lemma 10.5. Set \(\varvec{w}= \epsilon ^{-1} \nabla _{\varvec{\alpha }}\times \varvec{z}\), clearly \( \varvec{w}\in {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\). Then, from Theorem 5.7 and the inequality (10.13) there exists \(\varvec{w}_0 \in {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\) such that
Multiplying the first equation of (10.12) by \(\varvec{u}\) and integrating by parts, we obtain
Since \((\varvec{u},{\varvec{\nu }}) \in {\mathrm {Ker}(B_h)}\), we choose \(\psi _h\) as the \(L^2\)-projection of \(\psi \) in \(Q^{\varvec{\alpha }}_h\), then we have \(B_h(\varvec{u}, {\varvec{\nu }};\psi _h)=0\). Using the fact that \(\psi \in Q\) in the auxiliary problem (10.12) and \([[ \psi ]]_N=0\) on \({\mathcal {F}_h }\),
Using (10.14), we have
Using trace inequalities and (10.14), we have
Since \(\psi _h\) is the \(L^2\)-projection of \(\psi \), the third term is zero. Using (10.13), we obtain the following estimate for the last two terms:
From the results above, we have \(\Vert \varvec{u}\Vert _{0, \varOmega } \le C |( \varvec{u},{\varvec{\nu }}) |_{{\varvec{U}}(h)}\). \(\square \)
Appendix D: The Convergence of the Operator
Proof of Theorem 5.4
Let \((\varvec{u},p)\) be the analytical solution of (3.7), and \((\varvec{u}_h, {\varvec{\lambda }}_h, p_h)\) be the numerical solution of (5.4). By the triangle inequality and the definition of \( \Vert ( \cdot ,\cdot ) \Vert _{{\varvec{U}}(h)}\), we have
for any \((\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h\). First, we take \((\varvec{v},{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}\). Since \((\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \in {\mathrm {Ker}(B_h)}\), employing the ellipticity property of Theorem 5.3 and the definition of \(R^1_h\), we have
for any \(q\in Q^{\varvec{\alpha }}_h\). Combining (10.15) and (10.16), we have
Next, we prove that
Let \((\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h\), and consider the following problem: find \((\varvec{w},\nu ) \in {\varvec{U}}(h)\) such that
Problem (10.19) admits a solution in \({\varvec{U}}(h)\) that is unique up to elements in \({\mathrm {Ker}(B_h)}\). The discrete inf-sup condition of Theorem 5.2 guarantees the existence of a solution \((\varvec{w},\nu ) \in {\varvec{U}}(h)\) satisfying
where we have used the continuity of \(B_h(\cdot ,\cdot ;\cdot ) \), the definition of the norm \(\Vert ( \cdot ,\cdot ) \Vert _{{\varvec{U}}(h)}\), and the definition of \(\mathcal {R}^2_h(\cdot ) \). From (10.19), \(B_h(\varvec{w}+\varvec{v},\nu +{\varvec{\eta }};q) =0\), for any \(q\in Q^{\varvec{\alpha }}_h\), so that \((\varvec{w}+\varvec{v},\nu +{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}\). Therefore, since
for any \((\varvec{v},{\varvec{\eta }})\in {\varvec{U}}(h)\), taking into account (10.20), we obtain (10.18). This, together with (10.17), yields
where the constant C depends on \(a_1\), \(a_2\) and \(k_1\). Choosing \({\varvec{\eta }}=\varvec{{0}}\) gives the error bound for \((\varvec{u}-\varvec{u}_h,{\varvec{\lambda }}_h)\).
We now turn to the bound for \(p-p_h\). Again by the triangle inequality, we have
for any \(q \in Q^{\varvec{\alpha }}_h\). Since
for any \((\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}(h)\), the discrete inf-sup condition of \(B_h(\cdot ,\cdot ;\cdot ) \) gives
This, together with (10.21), gives a bound for \(p-p_h\). \(\square \)
Proof of Lemma 5.8
Let \((\varvec{u},p)\) be the analytical solution of (3.7). Let \({\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}\) be the \(L^2\)-projection onto \({\varvec{V}}^{\varvec{\alpha }}_h\). For \((\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h\),
Similarly, for \(q \in Q^{\varvec{\alpha }}_h\),
\(\square \)
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Lu, Z., Cesmelioglu, A., Van der Vegt, J.J.W. et al. Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals. J Sci Comput 70, 922–964 (2017). https://doi.org/10.1007/s10915-016-0270-1
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DOI: https://doi.org/10.1007/s10915-016-0270-1