$$\begin{aligned}
  \label{AA4}
  \J(\uh)  &   = \sum_{i=1}^\Nh \x_i \J(\vp_i)
  = \bcT \bx = \bcT \mA^{-1} \bb 
   = \by^{\T} \bb 
    = \sum_{i=1}^{\Nh} \y_i \ell_h(\vp_i)
    = \ell_h(\zh), $$ which exhibits the algebraic analogue of .

Similarly, as in let $\bxk$ and $\byk$ be the algebraic analogues of the approximations $\uhk$ and $\zhk$, respectively. The corresponding residuals of are given by $$\begin{aligned}
  \label{AA5}
  \brk := \bb - \mA \bxk =\mA (\bx - \bxk) \quad \mbox{ and } \quad
  \bsk := \bc - \mA^{\T} \byk = \mA^{\T} (\by -  \byk).\end{aligned}$$ By a standard manipulation, we derive the following correspondence between the discretization and its algebraic representation which will be used in next paragraphs. Similarly as in , we have $$\begin{aligned}
  \label{aa1}
  \J(\uhk)  & = \bcT \bxk, \qquad 
  \ell_h(\zhk)   = \bykT \bb.\end{aligned}$$ From and , we obtain $$\begin{aligned}
  \label{aa3}
  \ah(\uhk, \zhk ) & = \bykT \mA \bxk.\end{aligned}$$ Additionally, employing and , we derive $$\begin{aligned}
  \label{aa4}
  \ah(\uh - \uhk, \zh - \zhk ) & = (\by - \byk)^{\T} \mA (\bx - \bxk) = 
  \bskT \mA^{-1} \brk.\end{aligned}$$ Finally, using , , and –, we obtain $$\begin{aligned}
  \label{aa5}
  \etaAk & = \ell(\zhk) - \ah(\uhk ,\zhk)
  = \bykT \bb - \bykT \mA \bxk
  = \bykT \brk, \\
    \etaDAk & = 