A Definitions of the cosine and sine Operators
In term of the eigenpairs \(\left\{ \left( \lambda _i,\phi _i\right) \right\} _{i=1}^\infty \), the cosine and sine operators can be expressed as
$$\begin{aligned} \sin \left( A^\alpha t\right) u(t)= & {} \sum ^\infty _{i=1}\sin \left( \lambda _i^\alpha t\right) \left\langle u(t),\phi _{i}(x)\right\rangle \phi _{i}(x)\\= & {} \sum ^\infty _{i=1}\sum ^\infty _{j=1}(-1)^{j-1}\frac{\left( \lambda _i^\alpha t\right) ^{2j-1}}{(2j-1)!}\left\langle u(t),\phi _{i}(x)\right\rangle \phi _{i}(x) \end{aligned}$$
and
$$\begin{aligned} \cos \left( A^\alpha t\right) u(t)= & {} \sum ^\infty _{i=1}\cos \left( \lambda _i^\alpha t\right) \left\langle u(t),\phi _{i}(x)\right\rangle \phi _{i}(x)\\= & {} \sum ^\infty _{i=1}\sum ^\infty _{j=0}(-1)^{j}\frac{\left( \lambda _i^\alpha t\right) ^{2j}}{(2j)!}\left\langle u(t),\phi _{i}(x)\right\rangle \phi _{i}(x). \end{aligned}$$
B Simulation of Stochastic Integral for fBm
Suppose \(0\le t_1\le \dots \le t_{m}\le \dots \le t_{M}=T\) \((m=1,2,\dots , M-1)\) and the fixed sizes of the mesh \(\tau =t_{m+1}-t_{m}\). Let’s consider the following vector
$$\begin{aligned} Z=\left( \int ^{t_{1}}_{0}s\mathrm {d}\beta _H(s),\int ^{t_{2}}_{t_{1}}(s-t_1)\mathrm {d}\beta _H(s),\dots , \int ^{t_{M}}_{t_{M-1}}(s-t_{M-1})\mathrm {d}\beta _H(s)\right) . \end{aligned}$$
The stochastic integral \(\int ^{t_{m+1}}_{t_{m}}(s-t_m)\mathrm {d}\beta _H(s)\) is a Gaussian process with mean 0. The Cholesky method can be applied to stationary and non-stationary Gaussian processes. Thus, we use the Ckolesky method to simulate (5.10). The probability distribution of the vector Z is normal with mean 0 and the covariance matrix \(\varSigma \). Let \(\varSigma _{i,j}\) be the element of row i, column j of matrix \(\varSigma \). Then
$$\begin{aligned} \varSigma _{i,j}= & {} \mathrm {E}\left[ \int ^{t_{j+1}}_{t_j}(s-t_j)\mathrm {d}\beta _H(s)\int ^{t_{k+1}}_{t_{k}}(t-t_k)\mathrm {d}\beta _H(t)\right] . \end{aligned}$$
By using Lemma 2.2, for \(j>k\), we have
$$\begin{aligned}&\mathrm {E}&\left[ \int ^{t_{j+1}}_{t_j}(s-t_j)\mathrm {d}B_H(s)\int ^{t_{k+1}}_{t_{k}}(t-t_k)\mathrm {d}B_H(t)\right] \\= & {} H(2H-2)\int ^{t_{j+1}}_{t_j}\int ^{t_{k+1}}_{t_{k}}(s-t_j)(t-t_k)(s-t)^{2H-2}\mathrm {d}t\mathrm {d}s\\= & {} -\frac{\tau ^2}{2}(t_{j+1}-t_{k+1})^{2H}+\frac{\tau }{2(2H+1)}\left( (t_{j+1}-t_{k})^{2H+1}-(t_{j}-t_{k+1})^{2H+1}\right) \\&-\frac{1}{2(2H+1)(2H+2)}\left( (t_{j+1}-t_{k})^{2H+2}-2(t_{j}-t_{k})^{2H+2}+(t_{j}-t_{k+1})^{2H+2}\right) \\= & {} -\frac{\tau ^{2+2H}}{2}(j-k)^{2H}+\frac{\tau ^{2+2H}}{2(2H+1)}\left( (j+1-k)^{2H+1}-(j-k-1)^{2H+1}\right) \\&-\frac{\tau ^{2+2H}}{2(2H+1)(2H+2)}\left( (j+1-k)^{2H+2}-2(j-k)^{2H+2}+(j-k-1)^{2H+2}\right) . \end{aligned}$$
When \(j=k\),
$$\begin{aligned} \mathrm {E}\left[ \int ^{t_{j+1}}_{t_j}(s-t_j)\mathrm {d}B_H(s)\int ^{t_{j+1}}_{t_{j}}(t-t_j)\mathrm {d}B_H(t)\right] =\frac{\tau ^{2H+2}}{2H+2}. \end{aligned}$$
When \(\varSigma \) is a symmetric positive matrix, the covariance matrix \(\varSigma \) can be written as \(L(M)L(M)'\), where the matrix L(M) is lower triangular matrix and the matrix \(L(M)'\) is the transpose of L(M). Let \(V=(V_1,V_2,\dots ,V_M)\). The elements of the vector V are a sequence of independent and identically distributed standard normal random variables. Since \(Z=L(M)V\), then Z can be simulated. Let \(l_{i,j}\) be the element of row i, column j of matrix L(M). That is,
$$\begin{aligned} \varSigma _{i,j}=\sum ^j_{k=1}l_{i,k}l_{j,k}, \quad j\le i. \end{aligned}$$
As \(i=j=1\), we have \(l^2_{1,1}=\varSigma _{1,1}\). The \(l_{i,j}\) satisfies
$$\begin{aligned} l_{i+1,1}= & {} \frac{\varSigma _{i+1,1}}{l_{1,1}},\\ l^2_{i+1,i+1}= & {} \varSigma _{i+1,i+1}-\sum ^{i}_{k=1}l^2_{i+1,k},\\ l_{i+1,j}= & {} \frac{1}{l_{j,j}}\left( \varSigma _{i+1,j}-\sum ^{j-1}_{k=1}l_{i+1,k}l_{j,k}\right) ,\quad 1<j\le i. \end{aligned}$$
C Proof of Theorem 5.2
Proof
First, combining (4.7), (5.8), (5.11), the assumptions, and Corollary 2 leads to
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)} \lesssim \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\right. \nonumber \\&\left. \sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \left[ f_N\left( u^N(s)\right) -f_N\left( u^N(t_j)\right) -\int ^s_{t_j}f'_N\left( u^N(t_j)\right) v^N(t_j)\mathrm {d}r\right] \mathrm {d}s\right\| _{L^2(D,U)}\nonumber \\&~~~+\left\| \sum ^{m}_{j=1}A^{-\alpha }\frac{ \tau \cos \left( A^{\frac{\alpha }{2}}(t_{m+1}-t_{j+1})\right) -\int ^{t_{j+1}}_{t_j}\cos \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \mathrm {d}s}{\tau }\right. \nonumber \\&~~~\times \left. \left[ \tau f'_N\left( u^N(t_j)\right) v^N(t_j)-f_N\left( u^N_j\right) +f_N\left( u^N_{j-1}\right) \right] \right\| _{L^2(D,U)}\nonumber \\&~~~+\left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \left[ f_N\left( u^N(t_j)\right) -f_N\left( u^N_j\right) \right] \mathrm {d}s\right\| _{L^2(D,U)}\nonumber \\&~~~+\left\| \int ^{t_{1}}_{0}A^{-\frac{\alpha }{2}}\right. \nonumber \\&\left. \sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \left[ f_N\left( u^N(s)\right) -f_N\left( u^N_0\right) \right] \mathrm {d}s\right\| _{L^2(D,U)}+\tau ^{\min \left\{ \frac{\gamma }{\alpha },2\right\} }\left( \sum ^{N_1}_{i=0}\lambda _i ^{\min \{\gamma -\alpha ,\alpha \}-2\rho }\right) ^{\frac{1}{2}}\nonumber \\&\lesssim J_1+J_2+\sum ^{m}_{j=0}\tau \left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}+\tau ^2+\tau ^{\min \left\{ \frac{\gamma }{\alpha },2\right\} }\left( \sum ^{N_1}_{i=0}\lambda _i ^{\min \{\gamma -\alpha ,\alpha \}-2\rho }\right) ^{\frac{1}{2}}. \end{aligned}$$
(C.1)
When \(\frac{1}{2}<H<1\), let \(\theta =\min \left\{ \frac{\gamma -\alpha }{\alpha },1\right\} \). We have
$$\begin{aligned} J_1\lesssim & {} \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \\&\left. \times \left[ \int ^s_{t_j}f'_N\left( u^N(r)\right) v^N(r)\mathrm {d}r-\int ^s_{t_j}f'_N\left( u^N(t_j)\right) v^N(t_j)\mathrm {d}r\right] \mathrm {d}s\right\| _{L^2(D,U)}\\\lesssim & {} \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \\&\left. \times \int ^s_{t_j}\left( f'_N\left( u^N(r)\right) -f'_N\left( u^N(t_j)\right) \right) v^N(r)\mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\\&+\left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \\&\left. \times \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \left( \cos \left( A^{\frac{\alpha }{2}}(r-t_j)\right) -I\right) v^N(t_j)\mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\\&+\left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \\&\left. \times \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \left( v^N(r)-\cos \left( A^{\frac{\alpha }{2}}(r-t_j)\right) v^N(t_j)\right) \mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\\\lesssim & {} \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\left\| \left( u^N(r)-u^N(t_j)\right) \right. \\&\left. \times v^N(r)\right\| _{L^2(D,U)}\mathrm {d}r\mathrm {d}s+\sum ^{m}_{j=1}\tau ^\theta \int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\left\| A^{\frac{\theta \alpha }{2}}v^N(t_j)\right\| _{L^2(D,U)}\mathrm {d}r\mathrm {d}s+II\\\lesssim & {} \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\int ^r_{t_j}\left\| v^N(t)v^N(r)\right\| _{L^2(D,U)}\mathrm {d}t\mathrm {d}r\mathrm {d}\\&+\sum ^{m}_{j=1}\tau ^{2+\theta }\left\| A^{\frac{\theta \alpha }{2}}v^N(t_j)\right\| _{L^2(D,U)}+II\\\lesssim & {} \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\int ^r_{t_j}\left\| \left| v^N(t)\right| ^2+\left| v^N(r)\right| ^2\right\| _{L^2(D,U)}\mathrm {d}t\mathrm {d}r\mathrm {d}s\\&+\sum ^{m}_{j=1}\tau ^{2+\theta }\left\| A^{\frac{\theta \alpha }{2}}v^N(t_j)\right\| _{L^2(D,U)}+II\\\lesssim & {} \tau ^2\left( 1+\left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) \\&+\sum ^{m}_{j=1}\tau ^{2+\theta }\left\| A^{\frac{\theta \alpha }{2}}v^N(t_j)\right\| _{L^2(D,U)}+II. \end{aligned}$$
The condition \(\gamma >\alpha \) implies that \(\rho >\frac{d}{4}\). Then combining the fact that \(\{\beta ^{i}_{H}(t)\}_{i\in {\mathbb {N}}}\) are mutually independent, Lemma 2.2, and (4.10), we have
$$\begin{aligned} II= & {} \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \right. \\&\times \left. \left[ -A^{\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(r-t_j)\right) u^N(t_j)+\int ^{r}_{t_j}\cos \left( A^{\frac{\alpha }{2}}(r-y)\right) f_N\left( u^N(y)\right) \mathrm {d}y\right. \right. \\&+\left. \left. \int ^{r}_{t_j}\sum ^{N_1}_{i=1}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \sigma _i\phi _i(x)\mathrm {d}\beta _H^i(y)\right] \mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\\\lesssim & {} \tau ^{2+\theta }\sum ^{m}_{j=1}\left\| A^{\frac{\alpha (\theta +1)}{2}}u^N(t_j)\right\| _{L^2(D,U)}+\tau ^2\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right) \\&+\left( \sum ^{N_1}_{i=1}\lambda _i^{-2\rho }\mathrm {E}\left[ \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \right. \right. \right. \\&\times \left. \left. \left. \int ^{r}_{t_j}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \phi _i(x)\mathrm {d}\beta _H^i(y)\mathrm {d}r\mathrm {d}s\right\| ^2\right] \right) ^{\frac{1}{2}}\\\lesssim & {} \left( \sum ^{N_1}_{i=1}\lambda _i^{-2\rho }\mathrm {E}\left[ \int _D\sum ^{m}_{j=1}\sum ^{m}_{k=1}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\int ^{t_{k+1}}_{k_j}\int ^t_{k_j}\int ^{r}_{t_j}\int ^{r_1}_{t_k}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \right. \\&\times \left. \left. f'_N\left( u^N(t_j)\right) \cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \phi _i(x)A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-t)\right) f'_N\left( u^N(t_k)\right) \right. \right. \\&\times \left. \left. \cos \left( \lambda _i^{\frac{\alpha }{2}}(r_1-y_1)\right) \phi _i(x)|y-y_1|^{2H-2}\mathrm {d}y\mathrm {d}y_1\mathrm {d}r\mathrm {d}s\mathrm {d}r_1\mathrm {d}t\mathrm {d}x\right] \right) ^{\frac{1}{2}}\\&+\tau ^{2+\theta }\sum ^{m}_{j=1}\left\| A^{\frac{\alpha (\theta +1)}{2}}u^N(t_j)\right\| _{L^2(D,U)}\\\lesssim & {} \left( \tau ^4\sum ^{m}_{j=1}\sum ^{m}_{k=1}\mathrm {E}\left[ \left\| f'_N\left( u^N(t_j)\right) \phi _i(x)\right\| \left\| f'_N\left( u^N(t_k)\right) \phi _i(x)\right\| \right] \right. \\&\times \left. \int ^{t_{j+1}}_{t_j}\int ^{t_{k+1}}_{t_k}|y-y_1|^{2H-2}\mathrm {d}y\mathrm {d}y_1\right) ^{\frac{1}{2}}+\tau ^{2+\theta }\sum ^{m}_{j=1}\left\| A^{\frac{\alpha (\theta +1)}{2}}u^N(t_j)\right\| _{L^2(D,U)}. \end{aligned}$$
Combining the above estimates and Corollary 2 leads to
$$\begin{aligned} J_1\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\&+\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) , \ \alpha <\gamma \le 2\alpha \end{aligned}$$
and
$$\begin{aligned} J_1\lesssim & {} \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\&+\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) ,\ \gamma >2\alpha . \end{aligned}$$
Similar to \(J_1\), one gets
$$\begin{aligned} J_2\lesssim & {} \left\| \sum ^{m}_{j=1}\frac{ A^{-\frac{\alpha }{2}}\int ^{t_{j+1}}_{t_j}\int ^{t_{j+1}}_{s}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-r)\right) \mathrm {d}r\mathrm {d}s}{\tau }\left[ f_N\left( u^N(t_{j-1})\right) \right. \right. \\&-\left. \left. f_N\left( u^N(t_j)\right) +\tau f'_N\left( u^N(t_j)\right) v^N(t_j)\right] \right\| _{L^2(D,U)}\\&+\tau \sum ^{m}_{j=1}\left\| \left[ f_N\left( u^N(t_j)\right) -f_N\left( u^N(t_{j-1})\right) -f_N\left( u^N_j\right) +f_N\left( u^N_{j-1}\right) \right] \right\| _{L^2(D,U)}\\\lesssim & {} \left\| \sum ^{m}_{j=1}\frac{ A^{-\frac{\alpha }{2}}\int ^{t_{j+1}}_{t_j}\int ^{t_{j+1}}_{s}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-r)\right) \mathrm {d}r\mathrm {d}s}{\tau }\right. \\&\times \left. \left[ \int ^{t_j}_{t_{j-1}}f'_N\left( u^N(t_j)\right) v^N(t_j)\mathrm {d}r-\int ^{t_j}_{t_{j-1}}f'_N\left( u^N(r)\right) v^N(t_j)\mathrm {d}r\right] \right\| _{L^2(D,U)}\\&+\left\| \sum ^{m}_{j=1}\frac{ A^{-\frac{\alpha }{2}}\int ^{t_{j+1}}_{t_j}\int ^{t_{j+1}}_{s}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-r)\right) \mathrm {d}r\mathrm {d}s}{\tau }\right. \\&\times \left. \left[ \int ^{t_j}_{t_{j-1}}f'_N\left( u^N(r)\right) \left( v^N(t_j)-\cos \left( A^{\frac{\alpha }{2}}(t_{j}-r)\right) v^N(r)\right) \mathrm {d}r\right] \right\| _{L^2(D,U)}\\&+\left\| \sum ^{m}_{j=1}\frac{ A^{-\frac{\alpha }{2}}\int ^{t_{j+1}}_{t_j}\int ^{t_{j+1}}_{s}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-r)\right) \mathrm {d}r\mathrm {d}s}{\tau }\right. \\&\times \left. \left[ \int ^{t_j}_{t_{j-1}}f'_N\left( u^N(r)\right) \left( \cos \left( A^{\frac{\alpha }{2}}(t_{j}-r)\right) -I\right) v^N(r)\mathrm {d}r\right] \right\| _{L^2(D,U)}\\&+\tau \sum ^{m}_{j=0}\left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}. \end{aligned}$$
For \(\alpha <\gamma \le 2\alpha \), we have
$$\begin{aligned} J_2\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\+ & {} \left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) +\tau \sum ^{m}_{j=0}\left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}. \end{aligned}$$
When \(\gamma >2\alpha \), we also have
$$\begin{aligned} J_2\lesssim & {} \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\+ & {} \left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) +\tau \sum ^{m}_{j=0}\left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}. \end{aligned}$$
Combining (C.1), \(J_1\), and \(J_2\) leads to
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)}\\&\quad \lesssim \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\&\qquad +\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) ,\alpha <\gamma \le 2\alpha \end{aligned}$$
and
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)}\\&\quad \lesssim \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \\&\qquad +\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) , \gamma >2\alpha . \end{aligned}$$
When \(H=\frac{1}{2}\), using the same steps in (C.1), we get
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)}\\&\lesssim I_1+I_2+\sum ^{m}_{j=0}\tau \left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}+\tau ^2. \end{aligned}$$
For \(\alpha <\gamma \le 2\alpha \), using Corollary 2 and the assumptions of f, we obtain
$$\begin{aligned} I_1\lesssim & {} \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \nonumber \\&\times \left. \int ^s_{t_j}\left( f'_N\left( u^N(r)\right) -f'_N\left( u^N(t_j)\right) \right) v^N(r)\mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\nonumber \\&+\left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \nonumber \\&\times \left. \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \left( \cos \left( A^{\frac{\alpha }{2}}(r-t_j)\right) -I\right) v^N(t_j)\mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\nonumber \\&+\left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \right. \nonumber \\&\times \left. \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \left( v^N(r)-\cos \left( A^{\frac{\alpha }{2}}(r-t_j)\right) v^N(t_j)\right) \mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\nonumber \\\lesssim & {} \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\left\| \left( u^N(r)-u^N(t_j)\right) \times v^N(r)\right\| _{L^2(D,U)}\mathrm {d}r\mathrm {d}s\nonumber \\&+\sum ^{m}_{j=1}\tau ^{\frac{\gamma -\alpha }{\alpha }}\int ^{t_{j+1}}_{t_j}\int ^s_{t_j}\left\| A^{\frac{\gamma -\alpha }{2}}v^N(t_j)\right\| _{L^2(D,U)}\mathrm {d}r\mathrm {d}s+III\nonumber \\\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \nonumber \\&+\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) +III. \end{aligned}$$
(C.2)
Combining the fact that \(\{\beta ^{i}_{H}(t)\}_{i\in {\mathbb {N}}}\) are mutually independent and Equation (4.10), we have
$$\begin{aligned} III= & {} \left\| \sum ^{m}_{j=1}\int ^{t_{j+1}}_{t_j}A^{-\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \right. \\&\times \left. \left[ -A^{\frac{\alpha }{2}}\sin \left( A^{\frac{\alpha }{2}}(r-t_j)\right) u^N(t_j)+\int ^{r}_{t_j}\cos \left( A^{\frac{\alpha }{2}}(r-y)\right) f_N\left( u^N(y)\right) \mathrm {d}y\right. \right. \\&+\left. \left. \int ^{r}_{t_j}\sum ^{N_1}_{i=1}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \sigma _i\phi _i(x)\mathrm {d}\beta ^i(y)\right] \mathrm {d}r\mathrm {d}s\right\| _{L^2(D,U)}\\\lesssim & {} \tau ^{1+\frac{\gamma }{\alpha }}\sum ^{m}_{j=1}\left\| A^{\frac{\gamma }{2}}u^N(t_j)\right\| _{L^2(D,U)}+\tau ^2\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right) \\&+\left( \sum ^{m}_{j=1}\mathrm {E}\left[ \left\| \int ^{t_{j+1}}_{t_j}\sum ^{N_1}_{i=1}\lambda _i^{-\frac{\alpha }{2}}\sin \left( \lambda _i^{\frac{\alpha }{2}}(t_{m+1}-s)\right) \int ^s_{t_j}f'_N\left( u^N(t_j)\right) \right. \right. \right. \\&\times \left. \left. \left. \int ^{r}_{t_j}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \sigma _i\phi _i(x)\mathrm {d}\beta ^i(y)\mathrm {d}r\mathrm {d}s\right\| ^2\right] \right) ^{\frac{1}{2}}\\\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right) . \end{aligned}$$
In first inequality, we employ the fact that Brownian motion is a process with independent increment, that is
$$\begin{aligned}&\mathrm {E}&\left[ \int ^{t_{j+1}}_{t_j}\int ^s_{t_j}f'_N\left( u^N(t_j)\right) \int ^{r}_{t_j}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \mathrm {d}\beta ^i(y)\mathrm {d}r\mathrm {d}s\right. \\&\times \left. \int ^{t_{k+1}}_{t_k}\int ^s_{t_k}f'_N\left( u^N(t_k)\right) \int ^{r}_{t_k}\cos \left( \lambda _i^{\frac{\alpha }{2}}(r-y)\right) \mathrm {d}\beta ^i(y)\mathrm {d}r\mathrm {d}s\right] =0,\ j\ne k. \end{aligned}$$
Then
$$\begin{aligned} I_1\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}+\left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) . \end{aligned}$$
Similar to \(J_2\) and \(I_1\), we have
$$\begin{aligned} I_2\lesssim & {} \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}+\left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}\right. \\&+\left. \left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) +\tau \sum ^{m}_{j=0}\left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}. \end{aligned}$$
If \(\gamma >2\alpha \), then
$$\begin{aligned} I_1\lesssim & {} \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}+\left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) \end{aligned}$$
and
$$\begin{aligned} I_2\lesssim & {} \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}+\left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) \\&+\tau \sum ^{m}_{j=0}\left\| u^N(t_j)-u^N_j\right\| _{L^2(D,U)}. \end{aligned}$$
Using the above estimates and the discrete Grönwall inequality, we obtain
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)}\nonumber \\&\quad \lesssim \tau ^{\frac{\gamma }{\alpha }}\left( \frac{T^H}{\varepsilon }+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \nonumber \\&\qquad +\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) , \alpha <\gamma \le 2\alpha \end{aligned}$$
(C.3)
and
$$\begin{aligned}&\left\| u^N(t_{m+1})-u^N_{m+1}\right\| _{L^2(D,U)}\nonumber \\&\quad \lesssim \tau ^{2}\left( T^H+\left\| u_0\right\| _{L^2(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| _{L^2(D,{\dot{U}}^{\gamma -\alpha })}\right. \nonumber \\&\qquad +\left. \left\| u_0\right\| ^2_{L^4(D,{\dot{U}}^\gamma )}+\left\| v_0\right\| ^2_{L^4(D,{\dot{U}}^{\gamma -\alpha })}\right) ,\gamma >2\alpha . \end{aligned}$$
(C.4)
Take \(0<\tau <1\) and \(\varepsilon =\frac{1}{|\log (\tau )|}\). Combining above estimates and Theorem 4.1, we obtain the desired results. \(\square \)
