Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation

Abstract

We model the electrical activity of biological cells under external stimuli via a novel boundary integral (BI) formulation together with a suitable time-space numerical discretization scheme. Ionic channels follow a non-linear dynamic behavior commonly described by systems of ordinary differential equations dependent on the electric potential jump across the membrane. Since potentials in both intra– and extracellular domains satisfy an electrostatic approximation, we represent them using solely Dirichlet and Neumann traces over the membrane via boundary potentials. Hence, the volume problem is condensed to one posed over the cell boundary. A second-order time-stepping semi-implicit numerical Galerkin scheme is proposed and analyzed wherein BI operators are approximated by low-order basis functions, with stability independent of space discretization.

Appendices

Appendix A: Proof of Lemma 7

Assume $\phi \in \mathcal {C}^2[0,T]$, then for $t_1$, $t_2\in [0,T]$ and $t^{\star }:=\dfrac{t_1+t_2}{2}$ it holds

$$\begin{aligned} \frac{\phi (t_1)+\phi (t_2)}{2}-\phi (t^{\star }) = \frac{1}{2}\left[ \int _{t_1}^{t^{\star }}(s-t_1)\phi _{ss}ds - \int _{t^{\star }}^{t_2}(s-t_2)\phi _{ss}ds \right] . \end{aligned}$$

Then

$$\begin{aligned} \left| \frac{\phi (t_1)+\phi (t_2)}{2}-\phi (t^{\star }) \right|\le & {} \frac{1}{2}\left| \int _{t_1}^{t^{\star }}(s-t_1)ds\right| \max _{t\in [t_1,t^{\star }]} |\phi _{tt}(t)|\nonumber \\&+\,\frac{1}{2}\left| \int _{t^{\star }}^{t_2}(s-t_{2})ds\right| \max _{t\in [t^{\star },t_2]} |\phi _{tt}(t)|\\\le & {} \,\frac{1}{4}\left( (t_1-t_{\star })^2+(t_2-t_{\star })^2\right) \max _{t\in [t_1,t_2]} |\phi _{tt}(t)|.\nonumber \end{aligned}$$

(38)

For $u\in \mathcal {C}^2([0,T]; L^2(\Gamma ))$, from (38) and the definition of $\bar{u}^{n+\frac{1}{2}}$ it holds

$$\begin{aligned} \left\| \bar{u}^{n+\frac{1}{2}}-u^{n+\frac{1}{2}} \right\| _{L^2(\Gamma )} \le \frac{\tau ^2}{4} \max _{t\in [t_{n},t_{n+1}]} \left\| u_{tt}(t) \right\| _{L^2(\Gamma )} \end{aligned}$$

and

$$\begin{aligned} \left\| \hat{u}^{n+\frac{1}{2}}-\bar{u}^{n+\frac{1}{2}} \right\| _{L^2(\Gamma )} = \left\| \frac{u^{n+1}+u^{n-1}}{2}-u^n \right\| _{L^2(\Gamma )} \\ {}&\!\!\!\!\!\!\!\! \le \frac{\tau ^2}{16} \max \nolimits _{t\in [t_{n-1},t_{n+1}]} \left\| u_{tt}(t) \right\| _{L^2(\Gamma )}. \end{aligned}$$

By writing $u^{n+\frac{1}{2}}-\hat{u}^{n+\frac{1}{2}} = (u^{n+\frac{1}{2}}-\bar{u}^{n+\frac{1}{2}}) + (\bar{u}^{n+\frac{1}{2}}-\hat{u}^{n+\frac{1}{2}})$ together with the triangle inequality and (38) one can show that

$$\begin{aligned} \left\| u^{n+\frac{1}{2}}-\hat{u}^{n+\frac{1}{2}} \right\| _{L^2(\Gamma )} \le \frac{5}{16}\tau ^2 \max _{t\in [t_{n-1},t_{n+1}]} \left\| u_{tt}(t) \right\| _{L^2(\Gamma )}. \end{aligned}$$

If we further assume $\phi \in \mathcal {C}^3[0,T]$, then it holds

$$\begin{aligned} \frac{\phi (t_2)-\phi (t_1)}{\tau } - \phi _t(t^{\star }) = \frac{1}{2\tau }\left[ \int _{t_1}^{t^{\star }}(s-t_1)^2\phi _{sss}ds - \int _{t^{\star }}^{t_2}(s-t_2)\phi _{sss}ds \right] \end{aligned}$$

Thus, for $u\in \mathcal {C}^3([0,T]; L^2(\Gamma ))$, one proves

$$\begin{aligned} \left\| \bar{\partial }u^{n} - u^{n+\frac{1}{2}}_t \right\| _{L^2(\Gamma )} \le \frac{\tau ^2}{48} \max _{t\in [t_{n},t_{n+1}]} \left\| u_{ttt}(t) \right\| _{L^2(\Gamma )}, \end{aligned}$$

from where the result follows.

Appendix B: Proof of Lemma 10

For all $\varphi _h\in \mathcal {S}^1_h(\Gamma )$, we compute the discrete quantities:

$$\begin{aligned} c_m \left( \bar{\partial } \theta ^n,\varphi _h\right) _{\Gamma }+\mathsf {d}_{\Gamma }\left( \bar{\theta }^{n+\frac{1}{2}},\varphi _h\right)= & {} \ c_m \left( \bar{\partial } w^n_h,\varphi _h\right) _{\Gamma }+\mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h,\varphi _h\right) \\&-\,c_m \left( \bar{\partial } v^n_h,\varphi _h\right) _{\Gamma }-\mathsf {d}_{\Gamma }\left( \bar{v}^{n+\frac{1}{2}}_h,\varphi _h\right) \\= & {} \ c_m \left( \bar{\partial } w^n_h,\varphi _h\right) _{\Gamma }+\mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h,\varphi _h\right) \\&-\,\left\langle i^{n+\frac{1}{2}},\varphi _h \right\rangle + \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }},\varphi _h\right\rangle , \end{aligned}$$

then by adding a suitable zero, we write

$$\begin{aligned} c_m \left( \bar{\partial } \theta ^n,\varphi _h\right) _{\Gamma }+\mathsf {d}_{\Gamma }\left( \bar{\theta }^{n+\frac{1}{2}},\varphi _h\right) =&\ c_m\left( \bar{\partial } w^n_h-v^{n+\frac{1}{2}}_t,\varphi _h\right) _{\Gamma } +\mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h,\varphi _h\right) \\&+ c_m\left( v^{n+\frac{1}{2}}_t,\varphi _h\right) _{\Gamma }\\&-\left\langle i^{n+\frac{1}{2}},\varphi _h \right\rangle + \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }},\varphi _h\right\rangle . \end{aligned}$$

Then,

$$\begin{aligned} c_m (\bar{\partial } \theta ^n,\varphi _h)_{\Gamma }+\mathsf {d}_{\Gamma }\left( \bar{\theta }^{n+\frac{1}{2}},\varphi _h\right) =&\ c_m\left( \bar{\partial } w^n_h-v^{n+\frac{1}{2}}_t,\varphi _h\right) _{\Gamma } \\&+ \mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h,\varphi _h\right) - \mathsf {d}_{\Gamma }\left( v^{n+\frac{1}{2}},\varphi _h\right) \\&+\left\langle i^{n+\frac{1}{2}},\varphi _h \right\rangle - \left\langle i_{\text{ ion }}(v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}),\varphi _h\right\rangle \\&-\left\langle i^{n+\frac{1}{2}},\varphi _h \right\rangle +\left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }},\varphi _h \right\rangle . \end{aligned}$$

Choosing $\varphi _h = \bar{\theta }^{n+\frac{1}{2}}$ and by ellipticity of $\mathsf {d}_{\Gamma }(\cdot ,\cdot )$ (cf. Lemma 4), we get

$$\begin{aligned} \frac{c_m}{2}\bar{\partial }\left\| \theta ^n \right\| ^2_{L^2(\Gamma )} +\mu \left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )}\le & {} \ c_m\left| \left( \bar{\partial } w^n_h-v^{n+\frac{1}{2}}_t, \bar{\theta }^{n+\frac{1}{2}}\right) _{\Gamma }\right| \nonumber \\&+\left| \mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}}, \bar{\theta }^{n+\frac{1}{2}}\right) \right| \\&+ \left| \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}-i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) , \bar{\theta }^{n+\frac{1}{2}}\right\rangle \right| .\nonumber \end{aligned}$$

(39)

For the first term on the right hand side of (39), employ Young’s inequality for $\delta >0$, Lemmae 7 and 9 to obtain

$$\begin{aligned} \left| \left( \bar{\partial } w^n_h-v^{n+\frac{1}{2}}_t, \bar{\theta }^{n+\frac{1}{2}}\right) _{\Gamma }\right| \le&\, \frac{\delta }{2}\left\| \bar{\partial } w^n_h-v^{n+\frac{1}{2}}_t \right\| ^2_{L^2(\Gamma )} + \frac{1}{2\delta }\left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\ \le&\,\frac{\delta }{2}\left\| \bar{\partial }\rho ^{n+1} \right\| ^2_{L^2(\Gamma )} + \frac{\delta }{2}\left\| \bar{\partial }v^{n+\frac{1}{2}} - v^{n+\frac{1}{2}}_t \right\| ^2_{L^2(\Gamma )} + \frac{1}{2\delta }\left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\ \le&\, c\delta \left( h^2+\tau ^2\right) ^2 + \frac{1}{2\delta }\left\| {\bar{\theta }}^{n+1} \right\| ^2_{L^2(\Gamma )}. \end{aligned}$$

By (30), it holds $\mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}}, \varphi _h\right) =\mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h-w_h^{n+\frac{1}{2}}, \varphi _h\right)$ for all $\varphi _h\in \mathcal {S}^1_h(\Gamma )$. Then, for the second term on the right hand side in (39) by Young’s inequality for $\delta >0$, the continuity of $\mathsf {d}_{\Gamma }(\cdot ,\cdot )$, Lemmae 4 and 7 we get

$$\begin{aligned} \left| \mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}}, \bar{\theta }^{n+\frac{1}{2}}\right) \right|&= \left| \mathsf {d}_{\Gamma }\left( \bar{w}^{n+\frac{1}{2}}_h-w_h^{n+\frac{1}{2}}, \bar{\theta }^{n+\frac{1}{2}}\right) \right| \\&\le \alpha \left\| \bar{w}^{n+\frac{1}{2}}_h-w_h^{n+\frac{1}{2}} \right\| _{H^{\frac{1}{2}}(\Gamma )}\left\| \bar{\theta }^{n+\frac{1}{2}} \right\| _{H^{\frac{1}{2}}(\Gamma )}\\&\le \frac{\alpha \delta }{2}\left\| \bar{w}^{n+\frac{1}{2}}_h-w_h^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )}+\frac{\alpha }{2\delta }\left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )}\\&\le c\delta \tau ^4 + \frac{\alpha }{2\delta }\left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )} \end{aligned}$$

As for the last term on the right hand side of (39), we estimate

$$\begin{aligned}&\left| \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}-i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) ,\bar{\theta }^{n+\frac{1}{2}}\right\rangle \right| \le \left| \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}- \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) ,\bar{\theta }^{n+\frac{1}{2}}\right\rangle \right| \\&\quad + \left| \left\langle \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) -i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) ,\bar{\theta }^{n+\frac{1}{2}}\right\rangle \right| , \end{aligned}$$

where $\mathcal {I}_hi_{\text{ ion }}(v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}})$ is the nodal interpolation of the function $i_{\text{ ion }}(v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}})$. Again by Young’s inequality for $\delta >0$

$$\begin{aligned}&\left| \left\langle \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}-i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) ,\bar{\theta }^{n+\frac{1}{2}}\right\rangle \right| \le \frac{\delta }{2} \left\| \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}- \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) \right\| ^2_{L^2(\Gamma )}\\&\quad + \frac{\delta }{2} \left\| \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) -i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) \right\| ^2_{L^2(\Gamma )}\\&\quad + \frac{1}{\delta } \left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}. \end{aligned}$$

Then, by Lemma 5 we conclude

$$\begin{aligned} \left\| \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) -i_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) \right\| _{L^2(\Gamma )}\le c_{1}h^2. \end{aligned}$$

By (3), one derives

$$\begin{aligned}&\left\| \hat{\mathcal {I}}^{n+\frac{1}{2}}_{\text{ ion }}- \mathcal {I}_hi_{\text{ ion }}\left( v^{n+\frac{1}{2}},\mathbf{g}^{n+\frac{1}{2}}\right) \right\| ^2_{L^2(\Gamma )}\\&\quad \le 2 c^2_{\text{ ion }}\left( \left\| \hat{v}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )} + \left\| \hat{\mathbf{g}}^{n+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\right) . \end{aligned}$$

Putting all bounds together yields

$$\begin{aligned} \frac{c_m}{2}\bar{\partial }\left\| \theta ^n \right\| ^2_{L^2(\Gamma )} +\mu \left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )} \le&\ \frac{1}{\delta }\left( \frac{c_m}{2}+1\right) \left\| \bar{\theta }^n \right\| ^2_{L^2(\Gamma )} + \frac{\alpha }{2\delta } \left\| \bar{\theta }^{n+\frac{1}{2}} \right\| ^2_{H^{\frac{1}{2}}(\Gamma )}\\&+ c^2_{\text{ ion }}\delta \left\| \hat{v}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\&+ c^2_{\text{ ion }}\delta \left\| \hat{\mathbf{g}}^{n+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\&+ c\delta \left( h^2+\tau ^2\right) ^2. \end{aligned}$$

The continuous embedding of $H^{\frac{1}{2}}(\Gamma )$ into $L^2(\Gamma )$ means $\left\| u \right\| _{L^2(\Gamma )}\le c_{\Gamma }\left\| u \right\| _{H^{\frac{1}{2}}(\Gamma )}$ for all $u\in H^{\frac{1}{2}}(\Gamma )$ [51]. Then choosing $\delta = \frac{c^2_{\Gamma }}{\mu }\left( \frac{c_m}{2}+1\right) +\frac{\alpha }{2\mu }$ renders

$$\begin{aligned} \frac{c_m}{2}\bar{\partial }\left\| \theta ^n \right\| ^2_{L^2(\Gamma )} \le&\ c^2_{\text{ ion }}\delta \left\| \hat{v}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\&+ c^2_{\text{ ion }}\delta \left\| \hat{\mathbf{g}}^{n+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{n+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\\&+ c\delta \left( h^2+\tau ^2\right) . \end{aligned}$$

Observe that $\delta$ is independent of h and $\tau$. By (36), it holds

$$\begin{aligned} \hat{v}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}} = \theta ^{n} + \theta ^{n-1} + \rho ^n + \rho ^{n-1} + \hat{v}^{n+\frac{1}{2}}-v^{n+\frac{1}{2}}. \end{aligned}$$

Now, by Lemmae 7 and 9

$$\begin{aligned} \left\| \hat{v}^{n+\frac{1}{2}}_h-v^{n+\frac{1}{2}} \right\| _{L^2(\Gamma )} \le \left\| \theta ^{n} \right\| _{L^2(\Gamma )} + \left\| \theta ^{n-1} \right\| _{L^2(\Gamma )} + c\left( h^2+\tau ^2\right) \end{aligned}$$

and

$$\begin{aligned} \frac{c_m}{2}\bar{\partial }||\theta ^n||^2_{L^2(\Gamma )} \le&\ 2c^2_{\text{ ion }}\delta ||\theta ^{n}||^2_{L^2(\Gamma )} + 2c^2_{\text{ ion }}\delta ||\theta ^{n-1} ||^2_{L^2(\Gamma )}\\&+ c^2_{\text{ ion }}\delta ||\hat{\mathbf{g}}^{n+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{n+\frac{1}{2}}||^2_{L^2(\Gamma )} \\&+ c(h^2+\tau ^2). \end{aligned}$$

Reordering terms yields

$$\begin{aligned} \left\| \theta ^{n+1} \right\| ^2_{L^2(\Gamma )} + \tilde{\delta }\tau \left\| \theta ^{n} \right\| ^2_{L^2(\Gamma )} \le&\left( 1+2\tilde{\delta }\tau \right) \left( \left\| \theta ^{n} \right\| ^2_{L^2(\Gamma )} + \tilde{\delta }\tau \left\| \theta ^{n-1} \right\| ^2_{L^2(\Gamma )}\right) \\&+ \tilde{\delta }\tau ||\hat{\mathbf{g}}^{n+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{n+\frac{1}{2}}||^2_{L^2(\Gamma )} \\&+ c\tau \left( h^2+\tau ^2\right) ^2, \end{aligned}$$

where $\tilde{\delta } = 4\displaystyle \delta \frac{c^2_{\text{ ion }}}{c_m}$. Using the discrete version of Gronwall’s Lemma [44, Lemma 11.2] leads to

$$\begin{aligned} \left\| \theta ^{n+1} \right\| ^2_{L^2(\Gamma )} + \tilde{\delta }\tau \left\| \theta ^{n} \right\| ^2_{L^2(\Gamma )}\le & {} \ \left( 1+2\tilde{\delta }\tau \right) ^{n+1}\left( \left\| \theta ^{1} \right\| ^2_{L^2(\Gamma )}+ \tilde{\delta }\tau \left\| \theta ^{0} \right\| ^2_{L^2(\Gamma )} \right. \nonumber \\&\left. +\, \tilde{\delta }\tau \sum _{j=1}^{n}\left( 1+2\tilde{\delta }\tau \right) ^{-(j+1)}\left\| \hat{\mathbf{g}}^{j+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{j+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\right. \nonumber \\&+\left. c\tau \left( h^2+\tau ^2\right) ^2\sum _{j=1}^{n}\left( 1+2\tilde{\delta }\tau \right) ^{-(j+1)}\right) . \end{aligned}$$

(40)

Finally, by recalling

$$\begin{aligned} \sum _{j=1}^{n} \left( 1+2\tilde{\delta }\tau \right) ^{-(j+1)} = \frac{1}{2(1+2\tilde{\delta }\tau )\tilde{\delta }\tau }\left( 1 - \frac{1}{\left( 1+2\tilde{\delta }\tau \right) ^n}\right) \end{aligned}$$

together with $1+2\tilde{\delta }\tau \le \exp \left( 2\tilde{\delta }\tau \right)$ and $t_{n} = n\tau$,

$$\begin{aligned} \left\| \theta ^{n+1} \right\| ^2_{L^2(\Gamma )} \le&\ \exp \left( 2c_{\mathcal {HH}}t_{n+1}\right) \left( \left\| \theta ^{1} \right\| ^2_{L^2(\Gamma )}+ \tilde{\delta }\tau \left\| \theta ^{0} \right\| ^2_{L^2(\Gamma )} \right. \\&\left. + \tilde{\delta }\tau \sum _{j=1}^{n}\left( 1+2\tilde{\delta }\tau \right) ^{-(j+1)}\left\| \hat{\mathbf{g}}^{j+\frac{1}{2}}_h - \mathcal {I}_h \mathbf{g}^{j+\frac{1}{2}} \right\| ^2_{L^2(\Gamma )}\right) \\&+\frac{c}{2\tilde{\delta }}\left( h^2+\tau ^2\right) ^2\left( \exp \left( 2c_{\mathcal {HH}}t_{n}\right) -1\right) \end{aligned}$$

which is the first estimate stated in Lemma 10. For the second one, at each node $\mathbf{x}_m$, $m=1,\ldots , L$, it holds

$$\begin{aligned} \frac{d}{dt}{} \mathbf{g}^{n+\frac{1}{2}}_m = \mathcal {HH}\left( v^{n+\frac{1}{2}}_m,\mathbf{g}^{n+\frac{1}{2}}_m\right) \end{aligned}$$

(41)

where $v^{n+\frac{1}{2}}_m =v^{n+\frac{1}{2}}(\mathbf {x}_m)$ and $\mathbf{g}^{n+\frac{1}{2}}_m = \mathbf{g}^{n+\frac{1}{2}}(\mathbf {x}_m)$. From Taylor’s expansion around $t_{n+\frac{1}{2}}$, we obtain

$$\begin{aligned} \mathbf{g}^{n+1}_m&= \mathbf{g}^{n+\frac{1}{2}}_m+\frac{\tau }{2}\frac{d}{dt}\mathbf{g}^{n+\frac{1}{2}}_m + \frac{\tau ^2}{8}\frac{d^2}{dt^2}\mathbf{g}^{n+\frac{1}{2}}_m + \frac{\tau ^3}{48}\frac{d^3}{dt^3}\mathbf{g}^{\xi ^n_1}_m \end{aligned}$$

(42a)

$$\begin{aligned} \mathbf{g}^{n}_m&= \mathbf{g}^{n+\frac{1}{2}}_m-\frac{\tau }{2}\frac{d}{dt}\mathbf{g}^{n+\frac{1}{2}}_m + \frac{\tau ^2}{8}\frac{d^2}{dt^2}\mathbf{g}^{n+\frac{1}{2}}_m + \frac{\tau ^3}{48}\frac{d^3}{dt^3}\mathbf{g}^{\xi ^n_2}_m, \end{aligned}$$

(42b)

where $\xi ^n_1\in \left[ t_{n+\frac{1}{2}},t_{n+1}\right]$ and $\xi ^n_2\in \left[ t_n,t_{n+\frac{1}{2}}\right]$. Subtracting (42a) and (42b) and using (41) yields

$$\begin{aligned} \mathbf{g}^{n+1}_m - \mathbf{g}^{n}_m = \tau \mathcal {HH}\left( v^{n+\frac{1}{2}}_m,\mathbf{g}^{n+\frac{1}{2}}_m\right) + \frac{\tau ^3}{48}\left( \frac{d^3}{dt^3}{} \mathbf{g}^{\xi ^n_1}_m - \frac{d^3}{dt^3}{} \mathbf{g}^{\xi ^n_2}_m \right) . \end{aligned}$$

(43)

From Problem 5, at each node it holds

$$\begin{aligned} \bar{\partial }{} \mathbf{g}^n_{h,m}&= \mathcal {HH}\left( \hat{v}^{n+\frac{1}{2}}_{h,m},\hat{\mathbf{g}}^{n+\frac{1}{2}}_{h,m}\right) , \end{aligned}$$

for $m=1,\ldots ,L$, and subtraction with (43) delivers

$$\begin{aligned} \left( {\mathbf {g}}^{n+1}_{h,m} - {\mathbf {g}}^{n+1}_{m}\right) - \left( {\mathbf {g}}^{n}_{h,m} - {\mathbf {g}}^{n}_{m}\right) =&\ \tau \mathcal {HH}\left( \hat{v}^{n+\frac{1}{2}}_{h,m},\hat{\mathbf {g}}^{n+\frac{1}{2}}_{h,m}\right) - \tau \mathcal {HH}\left( v^{n+\frac{1}{2}}_m,{\mathbf {g}}^{n+\frac{1}{2}}_m\right) \\&- \frac{\tau ^3}{48}\left( \frac{d^3}{dt^3}{\mathbf {g}}^{\xi ^n_1}_m - \frac{d^3}{dt^3}{\mathbf {g}}^{\xi ^n_2}_m \right) . \end{aligned}$$

By (4) we have

$$\begin{aligned} \left| {\mathbf {g}}^{n+1}_{h,m} - {\mathbf {g}}^{n+1}_{m}\right| \le&\left| {\mathbf {g}}^{n}_{h,m} - {\mathbf {g}}^{n}_{m}\right| + c_{\mathcal {HH}}\tau \left| \hat{v}^{n+\frac{1}{2}}_{h,m} - v^{n+\frac{1}{2}}_m\right| \\&+ c_{\mathcal {HH}}\tau \left| \hat{\mathbf {g}}^{n+\frac{1}{2}}_{h,m} - {\mathbf {g}}^{n+\frac{1}{2}}_m\right| + \frac{\tau ^3}{48}\left| \frac{d^3}{dt^3}{\mathbf {g}}^{\xi ^n_1}_m - \frac{d^3}{dt^3}{\mathbf {g}}^{\xi ^n_2}_m \right| . \end{aligned}$$

Moreover, by Lemma 7, it holds

$$\begin{aligned} \left| \hat{\mathbf {g}}^{n+\frac{1}{2}}_{h,m} - {\mathbf {g}}^{n+\frac{1}{2}}_m\right| \le&\left| \hat{\mathbf {g}}^{n+\frac{1}{2}}_{h,m} - \hat{\mathbf { g}}^{n+\frac{1}{2}}_m\right| + \left| \hat{\mathbf {g}}^{n+\frac{1}{2}}_{m} - {\mathbf {g}}^{n+\frac{1}{2}}_m\right| \\ \le&\,\frac{3}{2}\left| {\mathbf {g}}^{n}_{h,m} - {\mathbf {g}}^{n}_m\right| + \frac{1}{2}\left| {\mathbf {g}}^{n-1}_{h,m} - {\mathbf {g}}^{n-1}_m\right| \\&+ \frac{5}{16}\tau ^2 \max _{t\in [t_{n-1},t_{n+1}]} \left| \frac{d^2}{dt^2}{\mathbf {g}}_m(t)\right| \end{aligned}$$

Then,

$$\begin{aligned} \left| \mathbf{g}^{n+1}_{h,m} - \mathbf{g}^{n+1}_{m}\right| \le&\left| \mathbf{g}^{n}_{h,m} - \mathbf{g}^{n}_{m}\right| + \frac{3}{2}c_{\mathcal {HH}}\tau \left| \mathbf{g}^{n}_{h,m} - \mathbf{g}^{n}_{m}\right| + \frac{1}{2}c_{\mathcal {HH}}\tau \left| \mathbf{g}^{n-1}_{h,m} - \mathbf{g}^{n-1}_{m}\right| \\&+ c_{\mathcal {HH}}\tau \left| \hat{v}^{n+\frac{1}{2}}_{h,m} - v^{n+\frac{1}{2}}_m\right| \\&+ \tau ^3\left( \frac{5}{16}c_{\mathcal {HH}} \max _{t\in [t_{n-1},t_{n+1}]} \left| \frac{d^2}{dt^2}\mathbf{g}_m(t)\right| + \frac{1}{48}\left| \frac{d^3}{dt^3}\mathbf{g}^{\xi ^n_1}_m - \frac{d^3}{dt^3}{} \mathbf{g}^{\xi ^n_2}_m \right| \right) \end{aligned}$$

Reordering terms and naming $\mathbf{e}^{n}_m = \mathbf{g}^{n}_{h,m} - \mathbf{g}^{n}_{m}$ we get

$$\begin{aligned}&\left| \mathbf{e}^{n+1}_m \right| + \frac{c_{\mathcal {HH}}}{2}\tau \left| \mathbf{e}^{n}_m \right| \le \left( 1+2c_{\mathcal {HH}}\tau \right) \left( \left| \mathbf{e}^{n}_m \right| + \frac{c_{\mathcal {HH}}}{2}\tau \left| \mathbf{e}^{n-1}_m \right| \right) \\&\quad +\, c_{\mathcal {HH}}\tau \left| \hat{v}^{n+\frac{1}{2}}_{h,m} - v^{n+\frac{1}{2}}_m\right| + \tilde{c} \tau ^3 \end{aligned}$$

where

$$\begin{aligned} \tilde{c} = \max _{n=1,\dots ,N-1}\left\{ \left( \frac{5}{16}c_{\mathcal {HH}} \max _{t\in [t_{n-1},t_{n+1}]} \left| \frac{d^2}{dt^2}{} \mathbf{g}_m(t)\right| + \frac{1}{48}\left| \frac{d^3}{dt^3}{} \mathbf{g}^{\xi ^n_1}_m - \frac{d^3}{dt^3}{} \mathbf{g}^{\xi ^n_2}_m \right| \right) \right\} \end{aligned}$$

Again, using the discrete version of Gronwall’s Lemma [44, Lemma 11.2]

$$\begin{aligned}&\left| \mathbf{e}^{n+1}_m\right| + \frac{c_{\mathcal {HH}}}{2}\tau \left| \mathbf{e}^{n}_m\right| \le \left( 1+2c_{\mathcal {HH}}\tau \right) ^{n+1}\left( \left| \mathbf{e}^{1}_m\right| + \frac{c_{\mathcal {HH}}}{2}\tau \left| \mathbf{e}^{0}_m\right| \right. \\&\quad \left. +\,c_{\mathcal {HH}}\tau \sum _{j=1}^{n}\left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)}\left| \hat{v}^{j+\frac{1}{2}}_{h,m} - v^{j+\frac{1}{2}}_m\right| \right. \\&\quad \left. +\,\tilde{c}\tau ^3 \sum _{j=1}^{n}\left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)}\right) . \end{aligned}$$

From the component-wise inequalities one derives estimates for entire $L^2(\Gamma )$-norm and one can conclude,

$$\begin{aligned} \left\| \mathbf{e}^{n+1} \right\| _{L^2(\Gamma )} \le \left( 1+2c_{\mathcal {HH}}\tau \right) ^{n+1}&\left( \left\| \mathbf{e}^{1} \right\| _{L^2(\Gamma )} + \frac{c_{\mathcal {HH}}}{2}\tau \left\| \mathbf{e}^{0} \right\| _{L^2(\Gamma )}\right. \\&\left. +\,c_{\mathcal {HH}}\tau \sum _{j=1}^{n}\left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)}\left\| \hat{v}^{j+\frac{1}{2}}_{h} - \mathcal {I}_hv^{j+\frac{1}{2}} \right\| _{L^2(\Gamma )}\right. \\&\left. +\,\tilde{c}\tau ^3 |\Gamma |\sum _{j=1}^{n}\left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)}\right) . \end{aligned}$$

Finally, recalling the basic geometric sum:

$$\begin{aligned} \sum _{j=1}^{n} \left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)} = \frac{1}{2(1+2c_{\mathcal {HH}}\tau )c_{\mathcal {HH}}\tau }\left( 1 - \frac{1}{\left( 1+2c_{\mathcal {HH}}\tau \right) ^n}\right) \end{aligned}$$

together with $1+2c_{\mathcal {HH}}\tau \le \exp \left( 2c_{\mathcal {HH}}\tau \right)$ and $t_{n} = n\tau$, one finally obtains

$$\begin{aligned} \left\| \mathbf{e}^{n+1} \right\| _{L^2(\Gamma )} \le&\exp \left( 2c_{\mathcal {HH}}t_{n+1}\right) \left( \left\| \mathbf{e}^{1} \right\| _{L^2(\Gamma )} + \frac{c_{\mathcal {HH}}}{2}\tau \left\| \mathbf{e}^{0} \right\| _{L^2(\Gamma )}\right) \\&+ \exp \left( 2c_{\mathcal {HH}}t_{n+1}\right) \left( c_{\mathcal {HH}}\tau \sum _{j=1}^{n}\left( 1+2c_{\mathcal {HH}}\tau \right) ^{-(j+1)}\left\| \hat{v}^{j+\frac{1}{2}}_{h} - \mathcal {I}_hv^{j+\frac{1}{2}} \right\| _{L^2(\Gamma )}\right) \\&+ |\Gamma | \frac{\tilde{c}}{2 c_{\mathcal {HH}}}\tau ^2 \left( \exp \left( 2c_{\mathcal {HH}}t_{n}\right) -1\right) . \end{aligned}$$

as stated.