### Numerical scheme

Assume that the computational domain Ω is a bounded polygonal set that is partitioned into *K* triangular elements \(T_{k}\) such that \(\overline{\Omega}= \bigcup_{k} T_{k}\). The finite element space is then taken to be \(V_{N}=\{v \in L^{2}(\Omega )^{3}: v|_{T_{k}} \in P_{N}(T_{k})^{3}\}\), where \(P_{N}(T_{k})\) denotes the space of polynomials of degree less than or equal to *N* defined on \(T_{k}\). On each element \(T_{k}\), the solution fields \({q}_{k}(x,y,t)= (E_{x}(x,y,t), E_{y}(x,y,t), H_{z}(x,y,t))^{T}\) are approximated by the piecewise polynomial functions \(\hat{q}_{k}(x,y,t) = (\hat{E}_{xk}(x,y,t),\hat{E}_{yk}(x,y,t), \hat{H}_{zk}(x,y,t))^{T}\).

In order to deduce the method, we start by multiplying equation (4) by test functions \(v \in V_{N}\), usually the Lagrange polynomials, and integrate over each element \(T_{k}\). The next step is to employ one integration by parts and to substitute in the resulting contour integral the flux *F* by a numerical flux \(F^{*}\). Reversing the integration by parts yields

$$\begin{aligned} \int_{T_{k}} \biggl(Q \frac{\partial\hat{q}}{\partial t} + \nabla\cdot F(\hat{q}) \biggr) \cdot v(x,y) \,dx \,dy = \int_{\partial T_{k}} \bigl(n \cdot \bigl(F(\hat{q})-F^{*}(\hat{q}) \bigr) \bigr) \cdot v(x,y) \,ds, \end{aligned}$$

where *n* is the outward pointing unit normal vector of the contour.

The approximate fields are allowed to be discontinuous across element boundaries. In this way, we introduce the notation for the jumps of the field values across the interfaces of the elements, \([\hat{E}]=\hat{E}^{-}-\hat{E}^{+}\) and \([\hat{H}]=\hat{H}^{-}-\hat{H}^{+}\), where the superscript “+” denotes the neighboring element and the superscript “−” refers to the local cell. Furthermore we introduce, respectively, the cell-impedances and cell-conductances \(Z^{\pm}=\mu^{\pm}c^{\pm}\) and \(Y^{\pm}= (Z^{\pm} )^{-1}\). At the outer cell boundaries we set \(Z^{+}=Z^{-}\).

The coupling between elements is introduced via numerical flux, defined by

$$\begin{aligned} n \cdot \bigl(F(\tilde{q})-F^{*}(\tilde{q}) \bigr) = \begin{pmatrix} \frac{-n_{y}}{Z^{+}+Z^{-}} (Z^{+} [\tilde{H}_{z}]-\alpha (n_{x}[\tilde{E}_{y}]-n_{y}[\tilde{E}_{x}] ) )\\ \frac{n_{x}}{Z^{+}+Z^{-}} (Z^{+} [\tilde{H}_{z}]-\alpha (n_{x}[\tilde{E}_{y}]-n_{y}[\tilde{E}_{x}] ) )\\ \frac{1}{Y^{+}+Y^{-}} (Y^{+} (n_{x}[\tilde{E}_{y}]-n_{y}[\tilde{E}_{x}] )-\alpha[\tilde{H}_{z}] ) \end{pmatrix}. \end{aligned}$$

The parameter \(\alpha\in[0,1]\) in the numerical flux can be used to control dissipation. Taking \(\alpha=0\) yields a non dissipative central flux while \(\alpha=1\) corresponds to the classic upwind flux.

To define the fully discrete scheme, we divide the time interval into *M* subintervals by the points \(0=t^{0}< t^{1}<\cdots<t^{M}=T\), where \(t^{m}=m \Delta t\), Δ*t* is the time step size and \(T+ \Delta t/2 \leq T_{f}\). The unknowns related to the electric field are approximated at integer time-stations \(t^{m}\) and are denoted by \(\hat{E}_{k}^{m}=\hat{E}_{k}(\cdot,t^{m})\). The unknowns related to the magnetic field are approximated at half-integer time-stations \(t^{m+1/2}=(m+ \frac{1}{2}) \Delta t\) and are denoted by \(\hat{H}_{k}^{m+1/2}=\hat{H}_{k}(\cdot,t^{m+1/2})\).

With the above setting, we can now formulate the iterative leap-frog DG method. The process starts with an approximation to the initial data which we denote by \((\hat{E}_{x}^{0}, \hat{E}_{y}^{0}, \hat{H}_{z}^{1/2})^{T} \in V_{N}\). For each \(m=0,1,\ldots, M-1\), we initialize the iterative process by

$$\hat{E}_{x}^{m+1,0}=\hat{E}_{x}^{m},\qquad \hat{E}_{y}^{m+1,0}=\hat{E}_{y}^{m},\qquad \hat{H}_{z}^{m+3/2,0}=\hat{H}_{z}^{m+1/2}. $$

The iterative scheme consists in calculating \((\hat{E}_{x}^{m+1},\hat{E}_{y}^{m+1}, \hat{H}_{z_{k}}^{m+3/2})^{T}\in V_{N}\), for all \((u, v, w)^{T} \in V_{N}\), in the following way. For \(n=0,1,2, \ldots\) , we find \(\hat{E}_{x_{k}}^{m+1,n+1}\) and \(\hat{E}_{y_{k}}^{m+1,n+1}\) such that

$$\begin{aligned} & \biggl(\epsilon_{xx} \frac{ \hat{E}_{x_{k}}^{m+1,n+1}-\hat{E}_{x_{k}}^{m}}{\Delta t} +\epsilon_{xy} \frac{ \hat{E}_{y_{k}}^{m+1,n+1}-\hat{E}_{y_{k}}^{m}}{\Delta t},u_{k} \biggr)_{T_{k}} \\ &\quad= \bigl( \partial_{y} \hat{H}_{z_{k}}^{m+1/2}, u_{k} \bigr) _{T_{k}} \\ &\qquad{} + \biggl(\frac{-n_{y}}{Z^{+}+Z^{-}} \bigl(Z^{+} \bigl[\hat{H}_{z}^{m+1/2} \bigr]-\alpha \bigl(n_{x}\bigl[\hat{E}_{y}^{[m+1/2,n]} \bigr]-n_{y}\bigl[\hat{E}_{x}^{[m+1/2,n]}\bigr] \bigr) \bigr),u_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(5)

$$\begin{aligned} & \biggl(\epsilon_{yx} \frac{ \hat{E}_{x_{k}}^{m+1,n+1}-\hat{E}_{x_{k}}^{m}}{\Delta t} + \epsilon_{yy} \frac{ \hat{E}_{y_{k}}^{m+1,n+1}-\hat{E}_{y_{k}}^{m}}{\Delta t}, v_{k} \biggr) _{T_{k} } \\ &\quad= - \bigl( \partial_{x} \hat{H}_{z_{k}}^{m+1/2}, v_{k} \bigr)_{T_{k}} \\ &\qquad{} + \biggl(\frac{n_{x}}{Z^{+}+Z^{-}} \bigl(Z^{+} \bigl[\hat{H}_{z}^{m+1/2} \bigr]-\alpha \bigl(n_{x}\bigl[\hat{E}_{y}^{[m+1/2,n]} \bigr]-n_{y}\bigl[\hat{E}_{x}^{[m+1/2,n]}\bigr] \bigr) \bigr), v_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(6)

until the stopping criterion \(\|\hat{E}^{m+1,n+1}-\hat{E}^{m+1,n}\|_{L^{2}(\Omega)}<\operatorname{tol}\) is satisfied for some pre-defined small constant tol. Then the correspondent numerical solution is denoted by \((\hat{E}_{x_{k}}^{m+1}, \hat{E}_{y_{k}}^{m+1})\). In the same way, we find \(\hat{H}_{z_{k}}^{m+3/2}\) by calculating \(\hat{H}_{z_{k}}^{m+3/2,n+1}\), for \(n=0,1,2, \ldots\) , such that

$$\begin{aligned} & \biggl( \mu\frac{ \hat{H}_{z_{k}}^{m+3/2,n+1}-\hat{H}_{z_{k}}^{m+1/2}}{\Delta t}, w_{k} \biggr)_{T_{k}} \\ &\quad = \bigl( \partial_{y} \hat{E}_{x_{k}}^{m+1} - \partial_{x} \hat{E}_{y_{k}}^{m+1},w_{k} \bigr)_{T_{k}} \\ &\qquad {}+ \biggl(\frac{1}{Y^{+}+Y^{-}} \bigl(Y^{+} \bigl(n_{x}\bigl[\hat{E}_{y}^{m+1}\bigr]-n_{y}\bigl[\hat{E}_{x}^{m+1}\bigr]\bigr)-\alpha\bigl[\hat{H}_{z}^{[m+1,n]} \bigr] \bigr), w_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(7)

until the stopping criterion \(\|\hat{H}^{m+1,n+1}-\hat{H}^{m+1,n}\|_{L^{2}(\Omega)}<\operatorname{tol}\) is satisfied. Here, \((\cdot,\cdot)_{T_{k}}\) and \((\cdot,\cdot)_{\partial T_{k}}\) denote the classical \(L^{2}(T_{k})\) and \(L^{2}(\partial T_{k})\) inner-products and \(\hat{E}^{[m+1/2,n]}\) and \(\hat{H}^{[m+1,n]}\) are the average approximations

$$\hat{E}^{[m+1/2,n]}= \frac{\hat{E}^{m}+\hat{E}^{m+1,n}}{2},\qquad \hat {H}^{[m+1,n]}= \frac{\hat{H}^{m+1/2}+\hat{H}^{m+3/2,n}}{2}. $$

The SM-ABC are discretised as in [1, 5], that is, for both upwind and central fluxes, consider \(\alpha=1\) for the numerical flux at the outer boundary and \([\tilde{E}_{x}]=\tilde{E}_{x}^{-}\), \([\tilde{E}_{y}]=\tilde{E}_{y}^{-}\) and \([\tilde{H}_{z}]=\tilde{H}_{z}^{-}\). We want to remark that for the case \(\alpha=0\), the inner iterations change the solution only in the triangles which have boundary edges, accordingly to the discretization of the SM-ABC.

This iterative scheme may be viewed as a predictor-corrector type method of the form P(C)^{niter}, where P represents the predictor step (the iteration \(n=0\)), C the correction steps (iterations \(n=1,2,\ldots \)), and niter the number of correction steps needed until the stopping criteria is verified. If we consider no inner iterations (\(\operatorname{niter}=0\)) we obtain the explicit method considered in [1].

### Convergence result

We will show that, under a suitable stability condition, the solution of the iterative predictor-corrector scheme (5)–(7) converges to the solution of the underlying implicit method. The implicit method is defined as follows: given an initial approximation \((\tilde{E}_{x}^{0}, \tilde{E}_{y}^{0}, \tilde{H}_{z}^{1/2})^{T} \in V_{N}\), for each \(m=0,1,\ldots, M-1\), we compute \((\tilde{E}_{x}^{m+1}, \tilde{E}_{y}^{m+1}, \tilde{H}_{z}^{m+3/2})^{T} \in V_{N}\) such that, for all \((u, v, w)^{T} \in V_{N}\),

$$\begin{aligned} & \biggl(\epsilon_{xx} \frac{ \tilde{E}_{x_{k}}^{m+1}-\tilde{E}_{x_{k}}^{m}}{\Delta t} +\epsilon_{xy} \frac{ \tilde{E}_{y_{k}}^{m+1}-\tilde{E}_{y_{k}}^{m}}{\Delta t},u_{k} \biggr)_{T_{k}} \\ &\quad= \bigl( \partial_{y} \tilde{H}_{z_{k}}^{m+1/2}, u_{k} \bigr) _{T_{k}} \\ &\qquad{} + \biggl(\frac{-n_{y}}{Z^{+}+Z^{-}} \bigl(Z^{+} \bigl[\tilde{H}_{z}^{m+1/2} \bigr]-\alpha \bigl(n_{x}\bigl[\tilde{E}_{y}^{[m+1/2]} \bigr]-n_{y}\bigl[\tilde{E}_{x}^{[m+1/2]}\bigr] \bigr) \bigr),u_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(8)

$$\begin{aligned} & \biggl(\epsilon_{yx} \frac{ \tilde{E}_{x_{k}}^{m+1}-\tilde{E}_{x_{k}}^{m}}{\Delta t} + \epsilon_{yy} \frac{ \tilde{E}_{y_{k}}^{m+1}-\tilde{E}_{y_{k}}^{m}}{\Delta t}, v_{k} \biggr) _{T_{k} } \\ &\quad = - \bigl( \partial_{x} \tilde{H}_{z_{k}}^{m+1/2}, v_{k} \bigr)_{T_{k}} \\ &\qquad{} + \biggl(\frac{n_{x}}{Z^{+}+Z^{-}} \bigl(Z^{+} \bigl[\tilde{H}_{z}^{m+1/2} \bigr]-\alpha \bigl(n_{x}\bigl[\tilde{E}_{y}^{[m+1/2]} \bigr]-n_{y}\bigl[\tilde{E}_{x}^{[m+1/2]}\bigr] \bigr) \bigr), v_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(9)

$$\begin{aligned} & \biggl( \mu\frac{ \tilde{H}_{z_{k}}^{m+3/2}-\tilde{H}_{z_{k}}^{m+1/2}}{\Delta t}, w_{k} \biggr)_{T_{k}} \\ &\quad= \bigl( \partial_{y} \tilde{E}_{x_{k}}^{m+1} - \partial_{x} \tilde{E}_{y_{k}}^{m+1},w_{k} \bigr)_{T_{k}} \\ &\qquad{} + \biggl(\frac{1}{Y^{+}+Y^{-}} \bigl(Y^{+} \bigl(n_{x}\bigl[\tilde{E}_{y}^{m+1}\bigr]-n_{y}\bigl[\tilde{E}_{x}^{m+1}\bigr]\bigr)-\alpha\bigl[\tilde{H}_{z}^{[m+1]}\bigr] \bigr), w_{k} \biggr)_{\partial T_{k}}, \end{aligned}$$

(10)

where we consider the average approximations \(\tilde{E}^{[m+1/2]}\) for \(\tilde{E}^{m+1/2}\) and \(\tilde{H}^{[m+1]}\) for \(\tilde{H}^{m+1}\) given by

$$ \tilde{E}^{[m+1/2]}=\frac{\tilde{E}^{m}+\tilde{E}^{m+1}}{2},\qquad \tilde{H}^{[m+1]}= \frac{\tilde{H}^{m+1/2}+\tilde{H}^{m+3/2}}{2}. $$

(11)

We note that the numerical solutions are defined implicitly, since the upwind fluxes involve the unknowns \(\tilde{E}_{x}^{m+1}\), \(\tilde{E}_{y}^{m+1}\) and \(\tilde{H}_{z}^{m+3/2}\).

Let \(h_{k}\) be the diameter of the triangle \(T_{k} \in\mathcal {T}_{h}\), and *h* be the maximum element diameter, that is, \(h_{k}=\sup_{P_{1},P_{2} \in T_{K}} \|P_{1}-P_{2}\|\), \(h=\max_{T_{k} \in\mathcal {T}_{h}}\{h_{k}\}\). We assume that the mesh is regular in the sense that there exists a constant \(\tau>0\) such that for all \(T_{k} \in\mathcal {T}_{h}\), \(\frac {h_{k}}{\tau_{k}} \leq\tau\), where \(\tau_{k}\) denotes the maximum diameter of a ball inscribed in \(T_{k}\). It may be proved (see [12]) that, for any \(u \in P_{N}(T_{k})\), the following trace inequality holds

$$ \Vert u \Vert _{L^{2}(f_{k})} \leq C_{\tau} \sqrt{(N+1) (N+2)} h_{k}^{-1/2} \Vert u \Vert _{L^{2}(T_{k})}, $$

(12)

where \(f_{k}\) is an edge of \(T_{k}\) and \(C_{\tau}\) a positive constant independent of \(h_{k}\) and *N* but dependent on the shape-regularity *τ*.

Let us now define the difference between two successive numeric values of the electromagnetic fields by

$$\begin{aligned} &\delta_{n} \hat{E}_{x_{k}}^{m+1} =\hat{E}_{x_{k}}^{m+1,n+1}-\hat{E}_{x_{k}}^{m+1,n}, \\ &\delta_{n} \hat{E}_{y_{k}}^{m+1} =\hat{E}_{y_{k}}^{m+1,n+1}-\hat{E}_{y_{k}}^{m+1,n}, \\ &\delta_{n} \hat{H}_{z_{k}}^{m+3/2} =\hat{H}_{z_{k}}^{m+3/2,n+1}-\hat{H}_{z_{k}}^{m+3/2,n}, \end{aligned}$$

for \(n = 0, 1, 2, \ldots \) .

We prove the following theorem by finding upper bounds for \(\delta_{n} \hat{E}_{x_{k}}^{m+1}\), \(\delta_{n} \hat{E}_{y_{k}}^{m+1}\) and \(\delta_{n} \hat{H}_{z_{k}}^{m+3/2}\).

### Theorem 1

*The solution of the iterative predictor*-*corrector scheme* (5)*–*(7) *converges to the solution of the method* (8)*–*(10) *provided that the stability condition of the underlying explicit method* (*i*.*e*., (5)*–*(7) *taking only the iteration*
\(n=0\)) *is satisfied*, *that is* (*see* [1])

$$ \Delta t < \frac{\min\{\underline{\epsilon}, \underline{\mu}\}}{\max \{ {C}_{E}, C_{H}\}} \min\{h_{k}\}, $$

(13)

*with*

$$\begin{aligned} &{C}_{E} = \frac{1}{2}C_{\mathrm{inv}}N^{2}+C_{\tau }^{2}(N+1) (N+2) \biggl(\frac{5}{2}+\frac{\alpha+\frac{1}{4}}{\min\{Z_{k}\}} \biggr), \\ &{C}_{H} = \frac{1}{2}C_{\mathrm{inv}}N^{2}+ C_{\tau}^{2}(N+1) (N+2) \biggl(\frac {5}{2}+ \frac{\alpha+\frac{1}{2}}{\min\{ Y_{k}\}} \biggr), \end{aligned}$$

*where*
\(C_{\tau}\)
*satisfies the trace inequality* (12), \(C_{\mathrm{inv}}\)
*is a positive constant independent of*
\(h_{k}\)
*and*
*N*, *and*
\(Z_{k}\)
*and*
\(Y_{k}\)
*denote respectively the cell*-*impedance*
*Z*
*and the cell*-*conductance*
*Y*
*inside the triangle*
\(T_{k} \in\mathcal {T}_{h}\).

### Proof

For simplicity of the prove, we consider that the mesh \(\mathcal {T}_{h}\) is conforming.

The stability condition (13) ensures that \(\|\delta_{0}\hat{E}^{m+1}\|_{L^{2}(\Omega)}\) and \(\|\delta _{0}\hat{H}_{z}^{m+3/2}\|_{L^{2}(\Omega)}\) are bounded for all \(m=0,1,\ldots,M-1\).

Let us denote by \(F^{\mathrm{int}}\) the set of internal edges and \(F^{\mathrm{ext}}\) the set of edges that belong to the boundary *δ*Ω. Let \(v_{k}\) be the set of indices of the neighbouring elements of \(T_{k}\). For each \(i\in v_{k}\), we consider the internal edge \(f_{ik} = T_{i}\cap T_{k}\), and we denote by \(n_{ik}\) the unit normal oriented from \(T_{i}\) towards \(T_{k}\). For each boundary edge \(f_{k} = T_{k} \cap\delta\Omega\), \(n_{k}\) is taken to be the unitary outer normal vector to \(f_{k}\).

Taking the difference of (5)–(7) between two successive iterations, \(n+1\) and *n*, and replacing \(u_{k}\), \(v_{k}\) and \(w_{k}\) by, respectively, \(\delta_{n} \hat{E}_{x_{k}}^{m+1}\), \(\delta_{n} \hat{E}_{y_{k}}^{m+1}\) and \(\delta_{n} \hat{H}_{z_{k}}^{m+3/2}\) and summing over all elements \(T_{k} \in\mathcal {T}_{h}\), we obtain

$$\begin{aligned} &\sum_{T_{k} \in\mathcal {T}_{h}} \bigl(\epsilon\delta_{n} \hat{E}_{k}^{m+1}, \delta_{n} \hat{E}_{k}^{m+1} \bigr)_{T_{k}} \\ &\quad= \frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl( \frac {(n_{y})_{ki}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ki} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{ki}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta _{n} \hat{E}_{x_{k}}^{m+1} \\ &\qquad{}+ \frac{(n_{y})_{ik}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ik} \delta_{n-1}\bigl[\hat{E}_{yi}^{m+1} \bigr]-(n_{y})_{ik}\delta_{n-1}\bigl[\hat{E}_{xi}^{m+1}\bigr] \bigr) \delta_{n} \hat{E}_{x_{i}}^{m+1} \biggr) \,ds \\ &\qquad{} -\frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl( \frac{(n_{x})_{ki}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ki} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{ki}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta_{n} \hat{E}_{y_{k}}^{m+1} \\ & \qquad{}+\frac{(n_{x})_{ik}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ik} \delta_{n-1}\bigl[\hat{E}_{yi}^{m+1} \bigr]-(n_{y})_{ik}\delta_{n-1}\bigl[\hat{E}_{xi}^{m+1}\bigr] \bigr) \delta_{n} \hat{E}_{y_{i}}^{m+1} \biggr) \,ds \\ &\qquad{}+\frac{ \Delta t}{2}\sum_{f_{k} \in F^{\mathrm{ext}} } \int_{f_{k}} \biggl(\frac {(n_{y})_{k}}{2 Z_{k}} \bigl((n_{x})_{k} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{k}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta_{n}\hat{E}_{x_{k}}^{m+1} \\ &\qquad{} -\frac{(n_{x})_{k}}{2 Z_{k}} \bigl((n_{x})_{k} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{k}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta_{n}\hat{E}_{y_{k}}^{m+1} \biggr) \,ds, \\ &\sum_{T_{k} \in\mathcal {T}_{h}} \bigl( \mu\delta_{n} \hat{H}_{z_{k}}^{m+3/2},\delta_{n} \hat{H}_{z_{k}}^{m+3/2} \bigr)_{T_{k}} \\ & \quad =-\frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl(\frac{1}{Y_{i}+Y_{k}} \delta_{n-1}\bigl[ \hat{H}_{z_{k}}^{m+3/2}\bigr]\delta_{n}\hat{H}_{z_{k}}^{m+3/2} \\ &\qquad{}+\frac{1}{Y_{i}+Y_{k}} \delta_{n-1}\bigl[ \hat{H}_{zi}^{m+3/2}\bigr]\delta_{n}\hat{H}_{zi}^{m+3/2} \biggr) \,ds \\ &\qquad{} -\frac{ \Delta t}{2}\sum_{f_{k} \in F^{\mathrm{ext}} } \int_{f_{k}} \biggl( \frac {1}{2 Y_{k}} \delta_{n-1} \bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \delta_{n}\hat{H}_{z_{k}}^{m+3/2} \biggr) \,ds. \end{aligned}$$

Then

$$\begin{aligned} &\sum_{T_{k} \in\mathcal {T}_{h}} \bigl(\epsilon\delta_{n} \hat{E}_{k}^{m+1}, \delta_{n} \hat{E}_{k}^{m+1} \bigr)_{T_{k}} \\ &\quad = \frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl( \frac {(n_{y})_{ki}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ki} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{ki}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta _{n} \bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \biggr) \,ds \\ & \qquad{}- \frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl( \frac{(n_{x})_{ki}}{Z_{i}+Z_{k}} \bigl((n_{x})_{ki} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{ki}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta_{n} \bigl[\hat{E}_{y_{k}}^{m+1}\bigr] \biggr) \,ds \\ & \qquad{}+\frac{ \Delta t}{2}\sum_{f_{k} \in F^{\mathrm{ext}} } \int_{f_{k}} \biggl(\frac {(n_{y})_{k}}{2 Z_{k}} \bigl((n_{x})_{k} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{k}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr) \delta_{n}\hat{E}_{x_{k}}^{m+1} \\ &\qquad{} -\frac{(n_{x})_{k}}{2 Z_{k}} \bigl((n_{x})_{k} \delta_{n-1}\bigl[\hat{E}_{y_{k}}^{m+1} \bigr]-(n_{y})_{k}\delta_{n-1}\bigl[\hat{E}_{x_{k}}^{m+1}\bigr] \bigr), \delta_{n}\hat{E}_{y_{k}}^{m+1} \biggr) \,ds, \end{aligned}$$

and

$$\begin{aligned} & \sum_{T_{k} \in\mathcal {T}_{h}} \bigl( \mu\delta_{n} \hat{H}_{z_{k}}^{m+3/2},\delta_{n} \hat{H}_{z_{k}}^{m+3/2} \bigr)_{T_{k}} \\ &\quad= -\frac{ \Delta t}{2}\sum_{f_{ik} \in F^{\mathrm{int}} } \int_{f_{ik}} \biggl(\frac {1}{Y_{i}+Y_{k}} \delta_{n-1}\bigl[ \hat{H}_{z_{k}}^{m+3/2}\bigr]\delta_{n}\bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \biggr) \,ds \\ &\qquad{} -\frac{ \Delta t}{2}\sum_{f_{k} \in F^{\mathrm{ext}} } \int_{f_{k}} \biggl( \frac {1}{2 Y_{k}} \delta_{n-1} \bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \delta_{n}\hat{H}_{z_{k}}^{m+3/2} \biggr) \,ds. \end{aligned}$$

So

$$\begin{aligned} &\sum_{T_{k} \in\mathcal {T}_{h}} \bigl(\epsilon\delta_{n} \hat{E}_{k}^{m+1}, \delta_{n} \hat{E}_{k}^{m+1} \bigr)_{T_{k}} \\ &\quad \leq \frac{ \Delta t}{4\min\{Z_{k}\}}\sum_{f_{ik} \in F^{\mathrm{int}} } \bigl\Vert \delta _{n-1}\bigl[\hat{E}_{k}^{m+1}\bigr] \bigr\Vert _{L^{2}(f_{ik})} \bigl\Vert \delta_{n} \bigl[\hat{E}_{k}^{m+1}\bigr] \bigr\Vert _{L^{2}(f_{ik})} \\ &\qquad{} +\frac{ \Delta t}{4\min\{Z_{k}\}}\sum_{f_{k} \in F^{\mathrm{ext}} } \bigl\Vert \delta_{n-1}\bigl[\hat{E}_{k}^{m+1}\bigr] \bigr\Vert _{L^{2}(f_{k})} \bigl\Vert \delta_{n} \bigl[\hat{E}_{k}^{m+1}\bigr] \bigr\Vert _{L^{2}(f_{k})}, \\ &\sum_{T_{k} \in\mathcal {T}_{h}} \bigl( \mu\delta_{n} \hat{H}_{z_{k}}^{m+3/2},\delta_{n} \hat{H}_{z_{k}}^{m+3/2} \bigr)_{T_{k}} \\ &\quad \leq \frac{ \Delta t}{4\min\{Y_{k}\}}\sum_{f_{ik} \in F^{\mathrm{int}} } \bigl\Vert \delta _{n-1}\bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \bigr\Vert _{L^{2}(f_{ik})} \bigl\Vert \delta_{n}\bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \bigr\Vert _{L^{2}(f_{ik})} \\ & \qquad{}+\frac{ \Delta t}{4\min\{Y_{k}\}}\sum_{f_{k} \in F^{\mathrm{ext}} } \bigl\Vert \delta_{n-1}\bigl[\hat{H}_{z_{k}}^{m+3/2}\bigr] \bigr\Vert _{L^{2}(f_{k})} \bigl\Vert \delta_{n}\hat{H}_{z_{k}}^{m+3/2} \bigr\Vert _{L^{2}(f_{k})}. \end{aligned}$$

Consequently, considering (12), we obtain

$$\begin{aligned} &\underline{\epsilon }\bigl\Vert \delta_{n} \hat{E}^{m+1} \bigr\Vert _{L^{2}(\Omega)} \\ & \quad\leq \frac{ \Delta t}{\min\{Z_{k}\}} C_{\tau}^{2} (N+1) (N+2)\max \bigl\{ h_{k}^{-1} \bigr\} \bigl\Vert \delta_{n-1}\hat{E}^{m+1} \bigr\Vert _{L^{2}(\Omega)}, \\ &\underline{\mu }\bigl\Vert \delta_{n} \hat{H}_{z_{k}}^{m+3/2} \bigr\Vert _{L^{2}(\Omega)} \\ &\quad \leq \frac{ \Delta t}{\min\{Y_{k}\}} C_{\tau}^{2} (N+1) (N+2)\max \bigl\{ h_{k}^{-1} \bigr\} \bigl\Vert \delta_{n-1}\hat{H}_{z}^{m+3/2} \bigr\Vert _{L^{2}(\Omega)}. \end{aligned}$$

Taking the following condition into account (that results from (13))

$$\Delta t < \frac{\min\{\underline{\epsilon}, \underline{\mu}\}\min\{Z_{k}, Y_{k}\}}{C_{\tau}^{2} (N+1)(N+2)} \min\{h_{k}\}, $$

we conclude that the sequences of general terms \(\| \delta_{n} \hat{E}^{m+1} \|_{L^{2}(\Omega)}\) and \(\| \delta_{n} \hat{H}_{z_{k}}^{m+3/2}\|_{L^{2}(\Omega)}\), \(n=0,1,\ldots\) , are decreasing and converge to 0. Then, the limit solution, as *n* tends to the infinity, of the iterative predictor-corrector scheme (5)–(7) converges to the unique solution of the method (8)–(10). □

From the proof of the previous theorem, if (13) is satisfied, we have that

$$\bigl\Vert \delta_{n} \hat{E}^{m+1} \bigr\Vert _{L^{2}(\Omega)} \leq (C_{*} )^{n} \bigl\Vert \delta_{0}\hat{E}^{m+1} \bigr\Vert _{L^{2}(\Omega)} $$

and

$$\bigl\Vert \delta_{n} \hat{H}_{z_{k}}^{m+3/2} \bigr\Vert _{L^{2}(\Omega)} \leq (C_{*} )^{n} \bigl\Vert \delta_{0}\hat{H}_{z}^{m+3/2} \bigr\Vert _{L^{2}(\Omega)}, $$

where \(C_{*}\) is dependent on the material properties *ϵ* and *μ*, on the properties of the space discretization namely *N* and *h* and on the time step Δ*t*,

$$C_{*} =\frac{\Delta t C_{\tau}^{2} (N+1)(N+2)}{\min\{\underline{\epsilon }, \underline{\mu}\}\min\{Z_{k}, Y_{k}\} \min\{h_{k}\}}< 1. $$

For a given *m*, it is possible to have an estimate for the number of inner iterations needed to satisfy the stopping criterion defined by tol, considering the approximation

$$C_{*} \simeq\max \biggl\{ \frac{ \Vert \delta_{1} \hat{E}^{m+1} \Vert _{L^{2}(\Omega)} }{ \Vert \delta_{0} \hat{E}^{m+1} \Vert _{L^{2}(\Omega)} }, \frac{ \Vert \delta_{1} \hat{H}_{z_{k}}^{m+3/2} \Vert _{L^{2}(\Omega)} }{ \Vert \delta _{0} \hat{H}_{z_{k}}^{m+3/2} \Vert _{L^{2}(\Omega)} } \biggr\} , $$

and so

$$\operatorname{niter} \simeq \bigl\lceil { \log\bigl(\operatorname{tol} \bigl\Vert \delta_{0}\hat{E}^{m+1} \bigr\Vert _{L^{2}(\Omega)}^{-1} \bigr)/\log(C_{*})} \bigr\rceil . $$

The next theorem establishes that the implicit method is second order convergent in time and arbitrary high order in space. So, with the previous result, we may conclude that, if a sufficient number of inner iterations is performed, i.e, if niter is sufficiently large, the iterative predictor-corrector scheme has, in practice, similar convergence properties.

### Theorem 2

*Let us consider the implicit leap*-*frog DG method* (8)*–*(10) *complemented with the discrete boundary conditions defined in Sect*. 2.1
*and suppose that the solution of the Maxwell’s equations* (1) *complemented by* (3) *has the following regularity*: \(E_{x}, E_{y}, H_{z} \in L^{\infty}(0,T_{f};H^{s+1}(\Omega))\), \(\frac{\partial E_{x}}{\partial t}, \frac{\partial E_{y}}{\partial t}, \frac{\partial H_{z}}{\partial t} \in L^{2} (0,T_{f};H^{s+1}(\Omega) \cap L^{\infty}(\partial \Omega))\)
*and*
\(\frac{\partial^{2} E_{x}}{\partial t^{2}}\), \(\frac {\partial^{2} E_{y}}{\partial t^{2}}\), \(\frac{\partial^{2} H_{z}}{\partial t^{2}}\) ∈ \(L^{2}(0,T_{f};H^{1}(\Omega))\), \(s \geq0\). *If the time step* Δ*t*
*satisfies*

$$ \Delta t < \frac{\min\{\underline{\epsilon}, \underline{\mu}\}}{\frac{1}{2} C_{\mathrm{inv}}N^{2} + 2C_{\tau}^{2}(N+1)(N+2)} \min\{h_{k}\}, $$

(14)

*where*
\(C_{\mathrm{inv}}\)
*and*
\(C_{\tau}\)
*are the positive constants defined in the previous theorem*, *then*

$$\begin{aligned} &\max_{1\leq m\leq M} \bigl( \bigl\Vert E^{m}-\tilde{E}^{m} \bigr\Vert _{L^{2}(\Omega)}+ \bigl\Vert H_{z}^{m+1/2}- \tilde{H}_{z}^{m+1/2} \bigr\Vert _{L^{2}(\Omega)} \bigr)\\ &\quad\leq C \bigl( \Delta t^{2} + h^{\min\{s,N\}}\bigr) \\ &\qquad{}+ C \bigl( \bigl\Vert E^{0}-\tilde{E}^{0} \bigr\Vert _{L^{2}(\Omega)}+ \bigl\Vert H_{z}^{1/2}-\tilde{H}_{z}^{1/2} \bigr\Vert _{L^{2}(\Omega)} \bigr) \end{aligned}$$

*holds*, *where*
*C*
*is a generic positive constant independent of* Δ*t*
*and the mesh size*
*h*.

### Proof

Follows the steps of the proof of Theorem 4.2 in [1]. □