$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Norms of mesh functions
- Problem: \( f^n =f(t_n) \) is a mesh function and hence not defined for all \( t \).
How to integrate \( f^n \)?
- Idea: Apply a numerical integration rule, using only
the mesh points of the mesh function.
The Trapezoidal rule:
$$ ||f^n|| = \left(\Delta t\left(\half(f^0)^2 + \half(f^{N_t})^2
+ \sum_{n=1}^{N_t-1} (f^n)^2\right)\right)^{1/2} $$
Common simplification yields the \( L^2 \) norm of a mesh function:
$$ ||f^n||_{\ell^2} = \left(\Delta t\sum_{n=0}^{N_t} (f^n)^2\right)^{1/2}$$
