estimates from the previous section, present the BiCG method allowing a simultaneous solution of the primal and dual problems and discuss several possibilities algebraic errors estimates and stopping criteria for the iterative solver.

Algebraic representation {#sec:alg1}
------------------------

Let $\{\vp_i:\Om\to\IR,\ i=1,\dots, \Nh \}$ be a basis of the finite-dimensional space $\Vh$. We define the matrix $\mA\in\IR^{\Nh\times\Nh}$ by $$\begin{aligned}
  \label{mA}
  \mA = \{\mA_{i,j}\}_{i,j=1}^{\Nh},\quad \mA_{i,j} := \ah(\vp_i,\vp_j),\ i,j=1,\dots, \Nh,\end{aligned}$$ where $\ah$ is the bilinear form defined by . Then the primal and dual discrete problems and are equivalent to the solution of two linear algebraic systems $$\begin{aligned}
  \label{AA1}
  \mA\bx = \bb\qquad\mbox{and}\qquad \mA^{\T}\by=\bc,\end{aligned}$$ respectively, where $\bx\in\IR^\Nh$ and $\by\in\IR^\Nh$ are the algebraic representation of the primal and dual solutions given by $$\begin{aligned}
  \label{AA2}
  & \bx = (\x_1,\dots, \x_{\Nh})^{\T} \ \leftrightarrow \uh =\sum\nolimits_{i=1}^\Nh \x_i \vp_i, \\
  \quad 
  \mbox{and} \quad &   \by = (\y_1,\dots, \y_{\Nh})^{\T} \ \leftrightarrow
  \zh =\sum\nolimits_{i=1}^\Nh \y_i \vp_i,
  \notag\end{aligned}$$ respectively, and $\bb\in\IR^\Nh$ and $\bc\in\IR^\Nh$ are the algebraic representation of the right-hand sides of the primal and dual problems given by $$\begin{aligned}
  \label{AA3}
  & \bb = (\b_1,\dots, \b_{\Nh})^{\T} \ \leftrightarrow \b_i = \ell_h(\vp_i),\ i=1,\dots,\Nh,
  \\ 
   \mbox{and} \quad
   &  \bc = (\c_1,\dots, \c_{\Nh})^{\T} \ \leftrightarrow \c_i =\J(\vp_i),\quad i=1,\dots,\Nh,
   \notag\end{aligned}$$ respectively. Using –, we obtain the equivalencies 