But the $n+1$ term bounds the error when Taylor series is within the radius of convergence, right? (The analogous condition holds for the example above)

My inclination was to think this works like taylor series approximation where the n+1 estimates the error. I suspect I'm wrong somehow though. I thought this might be correct because of the final inequality in the wiki article -- perhaps I could just use a n term taylor appx for $f_k$ and the actual square root for $f_l$: en.wikipedia.org/wiki/…

ah yes -- In this post I am considering a normalized $A$ in the expansion -- $A/\|A\|$ - which is 1/4 and 1 on the diagonal. So for this matrix I believe the spectrum is in the disk centered at 1 with radius 1.

I should have mentioned that the wiki article I was using for the formula is here: en.wikipedia.org/wiki/Square_root_of_a_matrix#Power_series -- the condition they have is $\|(I - A)\|^n \leq 1$ -- which is true for this case. Where are you seeing the eigenvalue condition you are referring too?

okay I'm almost there: for 1) Applying the poincare inequality gives: $$\|u_k - u^0\|_{L^1_{loc}} \leq \|Du_k - Du^0\|_{L^1_{loc}}$$ How do I get to $u_k-u^0$ are bounded in $L^2_{loc}$, I checked the relevant embedding theorem but it seems to be going the wrong way (Folland, Real Analysis 6.12)