Let we solved an equation (for example Poisson equation) by a numerical method, $$-\Delta u=f,~in~\Omega $$ $$u=g ,~on~ \partial \Omega $$
so we want to find the error ($u-u^h$),
we say a problem is unstable if a bit change in data leads to a large change in the solution, my question is:
If by a little change in the data, the answer be much better (unexpectedly), can we say the problem is unstable? For example if the rate of convergence is $2$ and for one step it is $60$.
Thanks for your help,
Stability means that the mapping from data $f$ to solution $u$ is continuous. Since this mapping is linear for linear problems, it is enough to show the existence of a constant $c>0$ such that $\|u\|\le c\|f\|$ for appropriate norms.
If for two different right-hand sides $f_1,f_2$ the solutions $u_1,u_2$ are much closer than expected, i.e., $\|u_1-u_2\|<<\|f_1-f_2\|$ this is not a contradiction to stability.
If on the other hand you can make $f_1-f_2$ arbitrarily small but $u_1-u_2$ does not vanish, this is a sign of instability.