\begin{align}
    \begin{split}
     \mathcal{C}(M) = &\: \Et^t(\theta,\S_t)^2
     + C_t M_t^{-\frac{2}{d}}
     + \Et^\pde(\theta,\S_\inte)^2 +C_\pde 
     M_\mathrm{int}^{-\frac{2}{d+1}}\\&+ C_1{T}^{\frac{1}{2}}
     \bigg[\Et^\divv(\theta,\S_\inte) +C_\divv 
     M_\mathrm{int}^{-\frac{1}{d+1}}+(1+\nu)
     big( \Et^s(\theta,\S_s) + C_s M_s^{-\frac{1}{d}}\big)\bigg],\\
    \end{split}
  \end{align}

  \begin{align}
    \begin{split}
    \norm{(u_0)_j-\hu_j(t=0)}_{L^2(\mathbb{T}^d)} &\leq 
    \norm{u_j-\hu_j}_{L^2(\partial \Omega)} \\&\leq 
    \sqrt{\frac{2\max\left\{2h_\Omega,d+1\right\}}
    {\rho_\Omega}}\norm{u_j-\hu_j}_{H^1(\Omega)}\\
    &\leq \sqrt{\frac{2\max\left\{2h_\Omega,d+1\right\}}
    {\rho_\Omega}}C_{1,k,d+1, u_1} \lambda_{1}(N)N^{-k+1}, 
    \end{split}
  \end{align}