> On 31 Dec 2020, at 12:09, 'David' via deal.II User Group
> <[email protected]> wrote:
>
>
> The absolute errors at the end of each iteration are of the order
> \mathcal{O}(10^{-10}). Also, if I compare my two-point residual assembly
> using matrix-free with the 'usual' spatial residual assembly and compute the
> l2 norm of the difference, i.e. l2_norm(r_mf - r-_tau), I obtain a difference
> of the order \mathcal{O}(10^{-10}). But from what I learnt, double precision
> should be at least accurate up to ~10^{-15}. There are five orders of
> magnitude in between. I probably miss something here in between. Any idea?
Well, you have to keep in mind that there are two more ingredients to keep into
account:
1. the Jacobian condition number (easily in the order of 10^3 — 10^6)
2. the number of operations that lead to the residuals (i.e., how many sums
your are eventually doing)
since errors accumulate, 10^{-10} is actually close to machine precision if you
have around 10^4/10^5 dofs.
Moreover, you are computing the l2_norm, i.e., sum over all dofs of the
*square* of the difference, and then you are taking the square root.
If you make 1e-15 error et each step in the square difference, propagate that
for ndofs sums, simply by taking the square root of the resulting error, you’d
get an expected error close to 1^{-8}, so you’re results are essentially
machine precision.
L.
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