An Efficient Spectral-Galerkin Approximation and Error Analysis for Maxwell Transmission Eigenvalue Problems in Spherical Geometries

Abstract

We propose and analyze an efficient spectral-Galerkin approximation for the Maxwell transmission eigenvalue problem in spherical geometry. Using a vector spherical harmonic expansion, we reduce the problem to a sequence of equivalent one-dimensional TE and TM modes that can be solved individually in parallel. For the TE mode, we derive associated generalized eigenvalue problems and corresponding pole conditions. Then we introduce weighted Sobolev spaces based on the pole condition and prove error estimates for the generalized eigenvalue problem. The TM mode is a coupled system with four unknown functions, which is challenging for numerical calculation. To handle it, we design an effective algorithm using Legendre-type vector basis functions. Finally, we provide some numerical experiments to validate our theoretical results and demonstrate the efficiency of the algorithms.

Appendices

Appendix A: Proof of Theorem 4

Let $u_*(t)=\frac{1}{4}(t+1)(1-t)^2\partial _tu(-1)$ for $\forall u\in H_{0,\omega ,l}^2(I)$. By construction, we have $\partial _t^k(u-u_*)(\pm 1)=0,(k=0,1).$ If $u\in H_{\omega ^{-2,-2},l}^s(I)$ , then we have $u-u_*\in H_{\omega ^{-2,-2},*}^s(I)$. In fact, from Hardy inequality (cf. B 8.8 in [17]) we have

$$\begin{aligned}&\int _I\omega ^{-2,-2}(u-u_*)^2dt\lesssim \int _I \left( \partial _t(u-u_*)\right) ^2dt,\\&\int _I\omega ^{-2,-2} \left( \partial _t(u-u_*)\right) ^2dt \lesssim \int _I\left( \partial _t^2(u-u_*)\right) ^2dt. \end{aligned}$$

Thus, we can derive that

$$\begin{aligned} \int _I\omega ^{-2,-2}(u-u_*)^2dt \lesssim \int _I\omega ^{-1,-1} \left( \partial _t(u-u_*)\right) ^2dt\lesssim \int _I \left( \partial _t^2(u-u_*)\right) ^2dt, \end{aligned}$$

Since

$$\begin{aligned} \int _I\left( \partial _t^2u_*\right) ^2dt&= \int _I\left( \left( \frac{3}{2}t-\frac{1}{2}\right) \partial _tu(-1)\right) ^2dt\\&=2\left( \partial _tu(-1)\right) ^2 =2\left( \int _I\partial _t^2udt\right) ^2\le 4\int _I\left( \partial ^2_tu\right) ^2dt, \end{aligned}$$

then we have

$$\begin{aligned} \int _I\left( \partial _t^2(u-u_*)\right) ^2dt \lesssim \int _I\left( \partial _t^2u\right) ^2dt+\int _I \left( \partial _t^2u_*\right) ^2dt \lesssim \int _I\left( \partial _t^2u\right) ^2dt. \end{aligned}$$

(A.1)

Similarly, we can derive that $\int _I\omega ^{1,1}(\partial _t^3u_*)^2dt=3(\partial _tu(-1))^2\lesssim \int _I(\partial ^2_tu)^2dt.$ Thus we have

$$\begin{aligned} \int _I\omega ^{1,1} \left( \partial _t^3(u-u_*)\right) ^2dt&\lesssim \int _I\omega ^{1,1} \left( \partial _t^3u\right) ^2dt+\int _I\omega ^{1,1} \left( \partial _t^3u_*\right) ^2dt \nonumber \\&\lesssim \int _I\omega ^{1,1} \left( \partial _t^3u\right) ^2dt +\int _I\left( \partial ^2_tu\right) ^2dt. \end{aligned}$$

(A.2)

For $k>3$, we have

$$\begin{aligned} \int _I\omega ^{-2+k,-2+k} \left( \partial _t^k(u-u_*)\right) ^2dt =\int _I\omega ^{-2+k,-2+k}\left( \partial _t^ku\right) ^2dt. \end{aligned}$$

(A.3)

Thus, $u-u_*\in H_{\omega ^{-2,-2},*}^s(I)$ and we can define

$$\begin{aligned} \pi _N^{2,l}u=\pi _{N,\omega ^{-2,-2}}(u-u_*)+u_* \in P_N^{0,l}, \forall u\in H_{\omega ^{-2,-2},l}^s(I) \end{aligned}$$

Then by Lemma 4.5 we can obtain

$$\begin{aligned} \Vert \partial _t^2\left( \pi _N^{2,l}u-u\right) \Vert =\Vert \partial _t^2\pi _{N,\omega ^{-2,-2}}(u-u_*)-(u-u_*)\Vert \lesssim N^{2-s}\Vert \partial _t^s(u-u_*)\Vert _{\omega ^{-2+s,-2+s}}. \end{aligned}$$

Together with (A.1), (A.2) and (A.3) we can get desired results. $\square$

Appendix B: Proof of Theorem 5

For any $v\in E_{1,m}$ it can be represented by $v=\sum _{i=1}^m\mu _iu_l^i$, we then have

$$\begin{aligned}&\frac{B(v,v)-B\left( \Pi _{N}^{2,l}v,\Pi _{N}^{2,l}v\right) }{B(v,v)}\le \frac{2|B\left( v,v-\Pi _{N}^{2,l}v\right) |}{B(v,v)}\\&\quad \le \frac{2\sum _{i,j=1}^{m}|\mu _i| |\mu _j||B\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^j\right) |}{\sum _{i=1}^{m}|\mu _i|^2}\\&\quad \le 2m\max _{i,j=1,\ldots ,m}|B\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^j\right) |. \end{aligned}$$

From Cauchy–Schwarz inequality we have

$$\begin{aligned}&\left| B\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^j\right) \right| =\frac{1}{\lambda ^j(\tau _l)}\left| \lambda ^j (\tau _l)B\left( u_l^j,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right| \\&\quad =\frac{1}{\lambda ^j(\tau _l)}\left| A_{\tau _l} \left( u_l^j,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right| =\frac{1}{\lambda ^j(\tau _l)}\left| A_{\tau _l} \left( u_l^j-\Pi _{N}^{2,l}u_l^j,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right| \\&\quad \le \frac{1}{\lambda ^j(\tau _l)} \left( A_{\tau _l}\left( u_l^j-\Pi _{N}^{2,l}u_l^j,u_l^j -\Pi _{N}^{2,l}u_l^j\right) \right) ^{\frac{1}{2}} \left( A_{\tau _l}\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right) ^{\frac{1}{2}}.\\ \end{aligned}$$

From Hardy inequality (cf. B 8.6 in [17]) we have

$$\begin{aligned} \int _I\frac{1}{(t+1)^2}u^2dt\lesssim \int _I\left( \partial _tu\right) ^2dt. \end{aligned}$$

Then from $\hbox {Poincar}\acute{e}$ inequality we can obtain

$$\begin{aligned} \Vert u\Vert _{2,\omega ,l}^2&=\int _I(t+1)^2 \left( \partial _t^2u\right) ^2dt+\int _I\left( \partial _tu\right) ^2dt +\int _I\frac{1}{(t+1)^2}u^2dt \\&\lesssim \int _I\left( \partial _t^2u\right) ^2dt +\int _I\left( \partial _tu\right) ^2dt\lesssim \int _I \left( \partial _t^2u\right) ^2dt. \end{aligned}$$

From the property of orthogonal project (4.34) and continuity of $A_{\tau _l}(u,v)$ in Theorem 1 we can derive that

$$\begin{aligned}&\left| B\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^j\right) \right| =\frac{1}{\lambda ^j(\tau _l)}\left| A_{\tau _l}\left( u_l^j,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right| \\&\le \frac{1}{\lambda ^j(\tau _l)} \left( A_{\tau _l}\left( u_l^j-\Pi _{N}^{2,l}u_l^j,u_l^j -\Pi _{N}^{2,l}u_l^j\right) \right) ^{\frac{1}{2}}\left( A_{\tau _l} \left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^i-\Pi _{N}^{2,l}u_l^i\right) \right) ^{\frac{1}{2}}\\&\le \frac{1}{\lambda ^j(\tau _l)} \left( A_{\tau _l}\left( u_l^j-\pi _{N}^{2,l}u_l^j,u_l^j-\pi _{N}^{2,l}u_l^j\right) \right) ^{\frac{1}{2}}\left( A_{\tau _l}\left( u_l^i-\pi _{N}^{2,l}u_l^i,u_l^i -\pi _{N}^{2,l}u_l^i\right) \right) ^{\frac{1}{2}}\\&\lesssim \frac{1}{\lambda ^j(\tau _l)}\Vert u_l^j-\pi _{N}^{2,l} u_l^j\Vert _{2,\omega ,l}\Vert u_l^i-\pi _{N}^{2,l}u_l^i\Vert _{2,\omega ,l}\\&\lesssim \frac{1}{\lambda ^j(\tau _l)} \Vert \partial _t^2\left( u_l^j-\pi _{N}^{2,l}u_l^j\right) \Vert \cdot \Vert \partial _t^2\left( u_l^i-\pi _{N}^{2,l}u_l^i\right) \Vert . \end{aligned}$$

Since

$$\begin{aligned} \frac{B(v,v)}{B\left( \Pi _{N}^{2,l}v,\Pi _{N}^{2,l}v\right) } \le \frac{1}{1-2m\max _{i,j=1,\ldots ,m}\left| B\left( u_l^i-\Pi _{N}^{2,l}u_l^i,u_l^j\right) \right| }, \end{aligned}$$

then from Theorems 3 and 4 we can get desired results.