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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}
