In [] (Theorem 6.1 p71) it is shown that the H1 error between the interpolate and the function is O(h) while the L2 error is O(h*h).
Take 3 triangulations by doubling each time the number of points on the boundary. Plot a function in the 3 cases, compute the L2 errors and plot them versus h. Check that they are aligned on a line of slope 1 in a log-log scale.