Convergence of an ADI splitting for Maxwell’s equations

Abstract

The convergence of an alternating direction implicit method for Maxwell’s equations on product domains is investigated. Unlike the classical Yee scheme and most other integrators proposed in the literature, this method is both unconditionally stable and computationally cheap. We prove second-order convergence of the time-discretization in the framework of operator semigroup theory. In contrast to formal considerations based on Taylor expansions, our convergence analysis respects the unboundedness of the involved differential operators. The proofs are based on results concerning the regularity of the Cauchy problems, which then allow to apply an abstract convergence proof by Hansen and Ostermann (Numer Math 108:557–570, 2008).

Appendix

We now present two proofs of Lemmas 3.6 and 3.7 we omitted in Sect. 3.

Proof of Lemma 3.6

Lax–Milgram provides us with a unique $v\in H^1_\Gamma (Q)$ satisfying (21). To show the asserted regularity of $v$, we consider the operators $A_j=-\partial _j^2$ on $L^2(Q)$ whose domain consists of those $w\in L^2(Q)$ such that $\partial _j^2 w\in L^2(Q)$, $w=0$ on $\Gamma _j^+$ or on $\Gamma _j^-$ if $\Gamma _j^+\subseteq \Gamma$ or if $\Gamma _j^-\subseteq \Gamma$, respectively, and $\partial _j w=0$ on $\Gamma _j^+$ or on $\Gamma _j^-$ if $\Gamma _j^+\subseteq \Gamma '$ or if $\Gamma _j^-\subseteq \Gamma '$, respectively. Here and below we have $j=1,2,3$. For $u\in D(A_j)$ and $v\in D_j$, an integration by parts shows

$$\begin{aligned} \int _Q (uv +A_ju \, v)\,\mathrm {d}x=\int _Q (uv+\partial _j u\,\partial _j v)\,\mathrm {d}x=:a(u,v), \end{aligned}$$

where $a$ is a symmetric, continuous and coercive bilinear form. It is routine to check that $A_j$ is the self adjoint operator induced by $a$. It is clear that $A_j$ is positive. In particular, $D_j$ is the domain of $A_j^\frac{1}{2}$ and hence $\partial _j A_j^{-\frac{1}{2}}$ is bounded on $L^2(Q)$.

To see that the resolvents of $A_i$ and $A_j$ commute, we observe that the resolvent of, say, $A_1$ is given by $((\lambda I+ A_1)^{-1}f)(x,y,z)= (R_1(\lambda )f(\cdot , y,z))(x)$ for $\lambda >0$, for almost every $(x,y,z)\in Q$ and the resolvent $R_1(\lambda )$ of the negative second derivative on $L^2(a_1^-,a_1^+)$ with the boundary conditions of $A_1$. Analogous facts hold for $A_2$ and $A_3$. If $f$ is the product of $f_k\in L^2(a_k^-,a_k^+)$ for $k=1,2,3$, then $(\lambda I+ A_i)^{-1}(\lambda I+ A_j)^{-1}f=(\lambda I+ A_j)^{-1}(\lambda I+ A_i)^{-1}f$. Since the span of such functions is dense in $L^2(Q)$, the resolvents commute.

As explained in Sects. III.4, VII.2 and X.1 of [21], we thus have a joint functional calculus with respect to $A_1$, $A_2$ and $A_3$ for Borel measurable functions $\phi :{\mathbb {R}}^3_+\rightarrow {\mathbb {R}}$. The operator $\phi (A_1,A_2,A_3)$ is bounded if $\phi$ is bounded, and for $h(\lambda )=1+\lambda _1+\lambda _2 +\lambda _3$ we have $h(A_1,A_2,A_3)=I+A_1+A_2+A_3=:I+A$ on the domain $D(A):=D(A_1)\cap D(A_2) \cap D(A_3)$. Set $\rho =1/h$. Then $\rho (A_1,A_2,A_3)$ is bounded and it is the inverse of $I+A$, so that $A$ is closed. Using the bounded functions $h_{i,j}(\lambda )=\lambda _i^\frac{1}{2}\,\lambda _j^\frac{1}{2} \rho (\lambda )$, we see that the operator $h_{ij}(A_1,A_2,A_3)=A_i^\frac{1}{2}\,A_j^\frac{1}{2}\, (I+A)^{-1}$ is bounded for all $i,j\in \{1,2,3\}$. This means that $D(A)\hookrightarrow H^2(Q)$ implying $D(A)=D$ and the equivalence of graph norm of $\Delta$ and the $H^2$-norm on $D$. It is then clear that $v=(I+A)^{-1}f$ is the required weak solution.$\square$

Proof of Lemma 3.7

1) Throughout, let $(\mathbf {E},\mathbf {H})\in D(M_0^2)$. It is known that a map $u\in H(\mathrm {rot})\cap H(\mathrm {div})$ belongs to $H^1(Q)^3$ if $u\times \nu =0$ or $u\cdot \nu =0$ holds on $\partial Q$. Moreover, the $H^1$ norm of $u$ is then dominated by $\Vert u\Vert _{L^2}+\Vert \mathrm{div }u\Vert _{L^2} +\Vert \mathrm{rot }u\Vert _{L^2}$, see, e.g., Theorem 2.17 in [1]. Note that the Eqs. (14) and (16) still hold on $Q$. In particular $\mathrm{div }\mathbf {E}$ and $\mathrm{div }\mathbf {H}$ belong to $L^2(Q)^3$. We thus have $\mathbf {E},\mathbf {H}\in H^1(Q)^3$ and $\Vert (\mathbf {E},\mathbf {H})\Vert _{H^1}\le c\, (\Vert (\mathbf {E},\mathbf {H})\Vert _X+\Vert M_0(\mathbf {E},\mathbf {H})\Vert _X)$. The asserted zero-order traces for $\mathbf {E}$ and $\mathbf {H}$ now are a direct consequence of the boundary conditions $\mathbf {E}\times \nu =0$ and $\mathbf {H}\cdot \nu =0$, respectively.

Since $\mathbf {E},\mathbf {H}\in H^1(Q)^3\hookrightarrow L^6(Q)^3$ and $M^2 (\mathbf {E},\mathbf {H})\in X$, Eq. (16) and the assumptions on $\varepsilon$ and $\mu$ imply that $\Delta E_j, \Delta H_j\in L^2(Q)$. A standard localization argument then yields $E_j,H_j\in H^2_{\text {loc}}(Q)^3$ for $j=1,2,3.$ In addition, the $X$-norm of $(\Delta \mathbf {E}, \Delta \mathbf {H})$ is bounded by that of $M_0^2(\mathbf {E},\mathbf {H})$ and $(\mathbf {E},\mathbf {H})$. We next establish the properties of the traces of $\mathbf {E}$ and $\mathbf {H}$ needed to derive $\mathbf {E},\mathbf {H}\in H^2(Q)^3$ from Lemma 3.6.

2) We first consider $E_1$. We will actually show that $\varepsilon E_1$ belongs to $H^2(Q)$ by applying Lemma 3.6 to $\varepsilon E_1$. Because of

$$\begin{aligned} \partial _{kl}E_1 = \frac{1}{\varepsilon } \,\partial _{kl}(\varepsilon E_1)- \frac{\partial _k \varepsilon }{\varepsilon }\, \partial _l E_1 - \frac{\partial _l \varepsilon }{\varepsilon }\, \partial _k E_1 -\frac{\partial _{kl} \varepsilon }{\varepsilon }\,E_1, \end{aligned}$$

(38)

it will then follow that $E_1\in H^2(Q)$ employing $E_1\in H^1(Q)$ and the assumed regularity of $\varepsilon$. At the present stage, from (38), $\Delta E_1\in L^2 (Q)$ and $E_1\in H^2_{\text {loc}}(Q)^3$ we can already infer that $f:=(I-\Delta )(\varepsilon E_1)\in L^2(Q)$ and $\varepsilon E_1\in H^2_{\mathrm {loc}}(Q)$. Part 1) shows that $\varepsilon E_1=0$ on the faces $\Gamma :=\Gamma _2^-\cup \Gamma _2^+\cup \Gamma _3^-\cup \Gamma _3^+$. Fix a function $\psi \in H^1(Q)$ with $\partial _2\psi ,\partial _3\psi \in H^1(Q)$ and having support in $[a_1^-,a_1^+]\times [a_2^-+\eta ,a_2^+-\eta ] \times [a_3^-+\eta ,a_3^+-\eta ]$ for some small $\eta =\eta (\psi )>0$. A given $\varphi \in H^1_\Gamma (Q)$ can be approximated in $H^1(Q)$ by such $\psi$ employing cutoff and mollification in the $(x_2,x_3)$ directions. For each sufficiently small $\kappa >0$, we set

$$\begin{aligned} Q_\kappa&=(a_1^-+\kappa ,a_1^+-\kappa )\times (a_2^-+\kappa ,a_2^+-\kappa ) \times (a_3^-+\kappa ,a_3^+-\kappa ). \end{aligned}$$

We take $\kappa \in (0,\eta (\psi ))$ and denote by $\Gamma _1^\pm (\kappa )$ the open faces of $Q_\kappa$ containing points of the form $(a_1^\pm \pm \kappa , x_2,x_3)$. Integrating by parts and using $\mathrm{div }(\varepsilon \mathbf {E})=0$ as well as $\partial _j(\varepsilon E_j)\in H^1_{\mathrm {loc}}(Q)$ for $j=1,2,3$, we conclude that

$$\begin{aligned}&\int _Q \nabla (\varepsilon E_1)\cdot \nabla \psi \,\mathrm {d}x +\int _Q \varepsilon E_1\psi \,\mathrm {d}x = \lim _{\kappa \rightarrow 0}\int _{Q_\kappa } \big (\varepsilon E_1\psi + \nabla (\varepsilon E_1)\cdot \nabla \psi \big ) \,\mathrm {d}x \nonumber \\&\quad \quad = \lim _{\kappa \rightarrow 0}\left[ \int _{Q_\kappa } (I-\Delta ) (\varepsilon E_1)\psi \,\mathrm {d}x +\int _{\partial Q_\kappa } \psi \,\nabla (\varepsilon E_1)\cdot \nu \,\mathrm {d}\sigma \right] \nonumber \\&\quad \quad = \int _{Q} f\psi \,\mathrm {d}x \pm \lim _{\kappa \rightarrow 0} \int _{\Gamma _1^\pm (\kappa )} \psi \,\partial _1 (\varepsilon E_1) \,\mathrm {d}(x_2,x_3)\nonumber \\&\quad \quad =\int _{Q} f\psi \,\mathrm {d}x \mp \lim _{\kappa \rightarrow 0} \int _{\Gamma _1^\pm (\kappa )} \psi (\partial _2 (\varepsilon E_2)+\partial _3(\varepsilon E_3)) \,\mathrm {d}(x_2,x_3)\nonumber \\&\quad \quad = \int _{Q} f\psi \,\mathrm {d}x \pm \lim _{\kappa \rightarrow 0} \int _{\Gamma _1^\pm (\kappa ) } \big ( \varepsilon E_2 \partial _2 \psi + \varepsilon E_3 \partial _3\psi \big ) \,\mathrm {d}(x_2,x_3)\nonumber \\&\quad \quad = \int _Q f\psi \,\mathrm {d}x. \end{aligned}$$

(39)

We have used that $\psi$ vanishes near $\Gamma$ for the penultimate equation and that $\varepsilon E_j, \partial _j \psi \in H^1(Q)^3$ and $\varepsilon E_j=0$ on $\Gamma _1^\pm$ for $j=2,3$ in the last identity, see part 1). By approximation, Eq. (39) then holds for all $\psi \in H^1_\Gamma (Q)$, and hence Lemma 3.6 yields $\varepsilon E_1\in H^2(Q)$ so that $E_1\in H^2(Q)$ as explained above. In the same way, one sees that $E_2,E_3\in H^2(Q)$. Moreover, $\Vert E_j\Vert _{H^2}$ is bounded by $c\,(\Vert E_j\Vert _{L^2} + \Vert \Delta E_j\Vert _{L^2})$ due to Lemma 3.6 and hence by $c\,(\Vert (\mathbf {E},\mathbf {H})\Vert _X+ \Vert M^2_0(\mathbf {E},\mathbf {H})\Vert _X)$ in view of step 1).

We denote by $\gamma _i$ the trace operator to $\Gamma _i^\pm$, where $i,j,k\in \{1,2,3\}$. Since $E_k\in H^2(Q)$, one can approximate $E_k$ in $H^2(Q)$ by $v_n\in C^2(\overline{Q})$. Clearly, $\gamma _i \partial _j v_n= \partial _j \gamma _i v_n$ and thus $\gamma _i\partial _j E_k =\partial _j \gamma _i E_k$. As a result, the asserted first order boundary conditions of $\mathbf {E}$ follow from the already established 0-order boundary conditions of $\mathbf {E}$.

3) Next, we consider $H_1$ and set $g:=(I-\Delta ) H_1\in L^2(Q)$. Here we have less Dirichlet boundary conditions, namely $H_j=0$ on $\Gamma _j^\pm$ for $j=1,2,3$. To deal with the Neumann conditions, we first note that

$$\begin{aligned}&\mathrm{rot }(\varepsilon ^{-1}\mathrm{rot }\mathbf {H})\in L^2(Q)^3, \qquad \varepsilon ^{-1}\mathrm{rot }\mathbf {H}\times \nu =0 \text {on}\partial Q,\\&\mathrm{div }(\varepsilon ^{-1}\mathrm{rot }\mathbf {H})= \nabla \varepsilon ^{-1}\cdot \mathrm{rot }\mathbf {H}\in L^2(Q). \end{aligned}$$

Hence, $\varepsilon ^{-1}\mathrm{rot }\mathbf {H}$ belongs to $H^1(Q)^3$ which yields $\mathrm{rot }\mathbf {H}\in H^1(Q)^3$. It also follows that $\mathrm{rot }\mathbf {H}\times \nu =0$ on $\partial Q$. In particular, the first component of $\mathrm{rot }\mathbf {H}$ vanishes on $\Gamma _2^\pm \cup \Gamma _3^\pm$.

We set $\tilde{\Gamma }= \Gamma _1 ^-\cup \Gamma _1^+$ and define the faces $\Gamma _j^\pm (\kappa )$ of $Q_\kappa$ in the $j$th direction for $j=2,3$, cf. step 2). We take functions $\psi \in H^1(Q)$ with $\partial _1\psi \in H^1(Q)$ and having support in $[a_1^-+\eta ,a_1^+-\eta ]\times [a_2^-,a_2^+] \times [a_3^-,a_3^+]$ for some $\eta >0$. We choose $\kappa \in (0,\eta )$ so that $\psi$ vanishes around $\Gamma _1^\pm (\kappa )$. As above, we deduce

$$\begin{aligned} \int _{Q}&\nabla H_1\cdot \nabla \psi \,\mathrm {d}x +\int _Q H_1\psi \,\mathrm {d}x = \lim _{\kappa \rightarrow 0} \int _{Q_\kappa } \big (H_1\psi +\nabla H_1\cdot \nabla \psi \big )\,\mathrm {d}x\\&= \lim _{\kappa \rightarrow 0}\Big [\int _{Q_\kappa } \psi \,(I-\Delta ) H_1\,\mathrm {d}x + \int _{\partial Q_\kappa } \psi \,\nu \cdot \nabla H_1\,\mathrm {d}\sigma \Big ]\\&=\int _{Q} \psi \,(I-\Delta ) H_1\,\mathrm {d}x + \lim _{\kappa \rightarrow 0} \int _{\partial Q_\kappa } \big [\psi \,\nu \cdot \nabla H_1 -(\mathrm{rot }\mathbf {H}\times \nu )\cdot (\psi ,0,0)\big ] \,\mathrm {d}\sigma \\&=\int _{Q} g\psi \,\mathrm {d}x + \lim _{\kappa \rightarrow 0} \int _{\partial Q_\kappa } \psi \,\nu \cdot \partial _1 \mathbf {H}\,\mathrm {d}\sigma \\&=\int _{Q} g\psi \,\mathrm {d}x \pm \lim _{\kappa \rightarrow 0}\Big [ \int _{\Gamma _2^\pm (\kappa )} \psi \, \partial _1 H_2 \,\mathrm {d}\sigma + \int _{\Gamma _3^\pm (\kappa )} \psi \, \partial _1 H_3 \,\mathrm {d}\sigma \Big ]\\&=\int _{Q} g\psi \,\mathrm {d}x \mp \lim _{\kappa \rightarrow 0}\Big [ \int _{\Gamma _2^\pm (\kappa )} H_2\, \partial _1 \psi \,\mathrm {d}\sigma + \int _{\Gamma _3^\pm (\kappa )} H_3\, \partial _1 \psi \,\mathrm {d}\sigma \Big ]\\&= \int _{Q} g\psi \,\mathrm {d}x. \end{aligned}$$

The remaining assertions now follow as in step 2.$\square$