I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the domain is discretized to m points; $x_i = x_0 + i.h$ where $h=1/(m-1)$ And the error is defined so: $E = U(x_i) - U_i$, for $i= 0,1,...,m-2$ and $u(0)=u(1)=0$ (as the BCs, for instance) which calculates the difference between the exact solution, $U(x_i)$, and the approximated, $U_i$.
My question is whether we should use function norm, or grid function norm, here or vector norm. My professor has stated that vector norms can not approximate functions which is correct and we should use here grid function norm. I want to know first why for calculating the global error we need these functions and how we can calculate them.
Thank you.
So this is a very delicate question and not ultimately sorted out - for hyperbolic problems for instance, the $L^1$ norm is the natural choice and the $L^\infty$ norm is for discontinuous problems relatively worthless. Overall, what norm you report (normalized or not, ...) depends strongly on the community you are reporting for.
If you are considering a continuous problem the $L^\infty$ norm is the strongest since you can show that also at e.g. boundaries nothing goes wrong - what you could actually hide in average norms such as $L^1$ and $L^2$.
I personally like relative $L^1$ and $L^\infty$ norms most, since they do not (in contrast to $L^2$ norms) cause values $<1$ to decay further and $>1$ to increase.