Consider the Eikonal equation \begin{align*} \begin{cases}\leftD u\right^{2}=1 & \text { on } \Omega \\ u \equiv 0 & \text { on } \partial \Omega\end{cases} \end{align*} and the viscous regularization \begin{align*} \begin{cases}\varepsilon \Delta u_{\varepsilon}+\leftD u_{\varepsilon}\right^{2}=1 & \text { on } \Omega \\ u_{\varepsilon} \equiv 0 & \text { on } \partial \Omega\end{cases} \end{align*} It is wellknown that $u_\varepsilon$ converges uniformly to the unique viscosity solution $u$ of the Eikonal equation. Does the error estimate $$\u_\varepsilon  u\_{\infty} \lesssim \sqrt{\varepsilon}$$ also holds?
1 Answer
Yes, this is a standard result. You can see sections 5.2 and 5.3 in these notes: https://wwwusers.cse.umn.edu/~jwcalder/viscosity_solutions.pdf
In fact, the rate is $O(\varepsilon)$ in one direction, when the boundary is $C^2$, since in this case $u$ is semiconcave.

$\begingroup$ In your notes, a lot of heavy lifting is one by the fact that the equation is of the form $u+H(x,\nabla u) = 0$, i.e. there's a zeroorder term appearing with the right sign. Is it actually possible to prove the same result for $H(x,\nabla u)=0$? $\endgroup$– ZylJul 7, 2022 at 22:26

$\begingroup$ Yes, have a look at Theorem 9.17 in my notes. This is for monotone finite difference schemes, but the proof is very similar to that of viscous approximations and this is for $H(x,\nabla u)=0$. You need some additional structure conditions on $H$ in this case to allow perturbation to strict sub/super solutions. $\endgroup$– JeffJul 8, 2022 at 0:29

$\begingroup$ The eikonal equations satisfies the structure condition I mentioned. $\endgroup$– JeffJul 8, 2022 at 0:30