Round-off errors relative to mesh size and polynomial order

Hello everyone,
I am conducting a convergence study on the L2 norm of the error using the ‘errornorm’ function for polynomial orders 1, 2, 3 and 4. I am noticing that as soon as the L2 error for polynomials of order 4 is expected to drop below 10^-11 on a given grid, I start losing convergence. Did anyone observe a similar behavior? Is there some empirical relationship between round-off errors, mesh size and polynomial order? I am running a 2D Poisson problem and I start observing this phenomenon for structured triangular grids larger than 256x256 elements.

Thank you in advance.

You’ve hit machine precision.

Thank you Nate. Do you know though of any back of the envelop calculation to determine the smallest sized grid that I run a simulation on given a certain polynomial order and a starting residual?

Thanks again.