Suppose I am solving a pde with a solution known with a finite-difference method. I can represent it as $A_hu_h=f$ for some approximating matrix $A_h$. And I define the discrete norm in which I will analyze the convergence to be $$||e||^2=h\sum e_i^2$$. I have the estimate for the error $$||e||\leq C ||\tau||$$ for $\tau$ local error if first order in space and $C$ a stability bound for $A^{-1}$. Then, I want to see which error gives my program to confirm the theory. I pick a point on the grid and watch what happens as a double the mesh. Assume I see the error gets decreased by 2, so I suspect it is a linear convergence at that point. Thus, I can see only rate of convergence at one point. How can I guarantee that the convergence is uniform "everywhere" on the grid? to confirm the theory results, should not I measure the error at each point and see I have a linear convergence in order to claim that the error goes to zero linearly in a discrete norm defined above? Because I could state results in a similar norms such as discrete $L^{\infty}$ or discrete $L^1$, however, what I measure by the computer is the same: difference between numerical solution and the function value at the point.
What confuses me is the fact that before I implement the method, I can state error estimates in a variety of norms, however, how do I relate these theoretical approximations to the error that I can actually measure, since this is just a value at the point on the grid and is independent from the way I do my theoretical analysis?
Edit: I should rephrase the question: There are a number of papers, where the convergence is measured for a particular point on the grid and the results are stated to confirm the theoretical estimates that are done in discrete $L^{\infty}$ or $L^{2}$ for the error vector. Is this sufficient to measure only at one point to obtain the confirmation and why the results presented only for the error at one point?
