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(I've removed my superfluous comments): I claimed earlier that the two sides scaled differently but they do not; @Hannes was kind enough to point out my error to me, thanks for that!
To build a bit upon what @almaz said: In one dimension, we have the embedding $W^{1,p}(\Omega) \subset C(\Omega)$. Point evaluations (i.e., integration w.r.t. a Dirac measure) are continuous functionals on $C(\Omega)$, regardless of the dimension. With the aforementioned embedding, in one dimension, they are also (well-defined) continuous functionals on $W^{1,p}(\Omega)$, so that their respective kernel is closed, and we can factor with respect to those kernels. The functional $f \mapsto \int_U f$ in well-defined and continuous on $W^{1,p}(\Omega)$ regardless of the dimension.
@KiraG. Fair enough. The equivalence between $p$-integrability of the gradient and difference quotients of a vector-valued $L^p(0,T,X)$ function can e.g. be found as Theorem 3.20 in M. Kreuter's master thesis (assuming that $X$ has the Radon-Nikodým property).
It gives an overview but also provides references for proofs. I've added the one that seems most important to me to my answer. Now, I have to admit that I don't see the explicit Euler scheme covered there right away, although other schemes are covered. It sounded to me like you were interested in having a starting point were not very much constrained on the precise method that is used. Does this help you?
