$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Abstract problem setting
Consider an abstract differential equation
$$ \mathcal{L}(u)=0\tp$$
Example: \( \mathcal{L}(u)=u'(t)+a(t)u(t)-b(t) \).
The corresponding discrete equation:
$$ \mathcal{L}_{\Delta}(u) =0\tp$$
Let now
- \( u \) be the numerical solution of the discrete equations,
computed at mesh points: \( u^n \), \( n=0,\ldots,N_t \)
- \( \uex \) the exact solution of the differential equation
$$
\begin{align*}
\mathcal{L}(\uex)&=0,\\
\mathcal{L}_\Delta(u)&=0\tp
\end{align*}
$$
\( u \) is computed at mesh points
