I'm currently solving the wave equation in a 1d and a 2d domain using the finite element method in space and the leapfrog or crank-nicolson method in time. Theoretically, I expect a convergence of $O(h^2 + \Delta t^2), (L2-norm)$ if the solution lies inside $C^4$. In 1d I get a convergence speed of ~2, as expected. In 2d my convergence speed is reduced to 1.7~1.8. Is this normal or a result of an implementation error? To calculate the error I calculate a reference solution on a finer net with more time steps.
2026-03-25 03:20:17.1774408817
How do theoretical convergence speeds translate into real life when using FEM?
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The order of error is only exact in the limit that the grid spacing goes to zero. For non-zero grid spacing we should expect to see some deviation from this. I would recommend performing the same simulation with a known exact solution at multiple (perhaps 5) different resolutions and plotting the error on log-log axis, along with a best fit line with the gradient $-2$. That way you'll be able to see if it's plausible that, in the limit, you get the expected convergence.