rate of convergence for the second order accurate method on two dimensional grid

Question

I am solving $u_t=u_{xx}+u_{yy}$ with a solver which is locally $O(dx^2+dy^2+dt^2)$. I use the following norm for the error between the numerical vector solution and the analytical solution $$||e||_2=\sqrt{dxdy\sum_{i,j}^{N_1,N_2}e_{ij}}=\sqrt{dxdyN_1,N_2O(dt^2+dx^2+dy^2)}$$ Now I want to test the rate of convergence, that is I increase the number of points in all dimensions by a factor of $2$ and thus new mesh sizes become $dx/2,dy/2,dt/2$ then the new norm of the error is $$||\bar{e}||:=\sqrt{dxdy/4*2N_12N_2O(dx^2/4+dy^2/4+dt^2/4)}=1/2||e||$$. So for the second order method discrete $L^2$ norm decreases by $2$, not by $4$? Even though at each point the local error will decrease by $4$. Did I do something wrong this is expected? I thought the total $L^2$ discrete norm is like averaged norm at each point, so the average should decrease by $4$ as well?

Answer 1

Yes, you've done something wrong. The appropriate 2-norm of a vector that approximates a grid function is

$$\|e\|_2 = \sqrt{dx dx \sum_{i,j} e_{ij}^2}.$$

You are incorrectly using $e_{ij}$ instead of $e_{ij}^2$.