Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\Omega_\delta$ is bounded by a constant which is independent of $\delta$ and somehow depending only on the original $\Omega$.

How can this be proved? I guess it is a classical result in numerical analysis, but I can't find a reference.

My motivation is this: suppose that we prove a discrete Poincaré inequality in $\Omega_\delta$. Then I want to get out of it the "continuous" Poincaré inequality without the measure of the domain blowing up in the limit. Does it make sense?

3more comments