Piecewise-constant discrete Aubin–Lions compactness

View each node in $V^{\ast }$ through the canonical embeddings $V\to H\to V^{\ast }$, and let ${\widehat{U}}_k$ be their continuous affine interpolation. On $I_n^k$ its derivative equals $D_k$. For $0\le s<T$ and $0<t<T-s$, the fundamental theorem for the affine function gives ${\widehat{U}}_k(t+s)-{\widehat{U}}_k(t)={\int }_t^{t+s}{D}_k(r)dr$. Minkowski [F2] and translation invariance of the time integral yield $\mathrm{\parallel }{\widehat{U}}_k(\cdot +s)-{\widehat{U}}_k{\mathrm{\parallel }}_{L^q(0,T-s;V^{\ast })}\le s\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^q(V^{\ast })}$.

If $t\in I_n^k$, write $t=t_{n-1}^k+\theta {\tau }_n^k$ with $0<\theta \le 1$. Then ${\bar{U}}_k(t)-{\widehat{U}}_k(t)=(1-\theta )(U_k^n-U_k^{n-1})$. Consequently, for $1<q<\mathrm{\infty }$, $\mathrm{\parallel }{\bar{U}}_k-{\widehat{U}}_k{\mathrm{\parallel }}_{L^q(V^{\ast })}^q\le {\sum }_n{\tau }_n^k\mathrm{\parallel }{U}_k^n-U_k^{n-1}{\mathrm{\parallel }}_{V^{\ast }}^q\le {\delta }_k^q\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^q(V^{\ast })}^q\mathrm{.}$ For $q=\mathrm{\infty }$, the same pointwise identity gives $\mathrm{\parallel }{\bar{U}}_k(t)-{\widehat{U}}_k(t){\mathrm{\parallel }}_{V^{\ast }}\le {\delta }_k\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(V^{\ast })}$ and hence $\mathrm{\parallel }{\bar{U}}_k-{\widehat{U}}_k{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(V^{\ast })}\le {\delta }_k\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(V^{\ast })}$. Combining the corresponding estimate at $t$ and $t+s$ with step 1.1 proves the stated $(s+2{\delta }_k)$ translate bound in both cases.

Fix $\epsilon >0$. By [F1], the image in $H$ of the $V$ unit ball has compact closure. Choose finitely many centers from that image forming an $\epsilon$-net, let $E\subset i(V)$ be their span, and let $P_E$ be the $H$-orthogonal projection. Its best-approximation property gives $\mathrm{\parallel }(I-P_E)iv{\mathrm{\parallel }}_H\le \epsilon \mathrm{\parallel }v{\mathrm{\parallel }}_V$ for every $v\in V$. Choose an $H$-orthonormal basis $e_j=i{v}_j$ of $E$. The scalar coefficients $a_{k,j}^n=(U_k^n,e_j{)}_H$ have weighted $L^p$ bounds from the $L^p(V)$ bound, and their total variation satisfies ${\sum }_n\mathrm{\mid }{a}_{k,j}^n-a_{k,j}^{n-1}\mathrm{\mid }\le \mathrm{\parallel }{v}_j{\mathrm{\parallel }}_V{\sum }_n{\tau }_n^k\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{V^{\ast }}\le T^{1-1\mathrm{/}q}\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_V\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^q(V^{\ast })}\mathrm{.}$ At least one node value is no larger than its weighted $L^p$ average, so the total-variation bound gives a uniform bound for every nodal coefficient.

The affine scalar interpolants of these finitely many coefficients are uniformly bounded; their derivatives are the pairings of $D_k$ with $v_j$, so their $L^q$ norms are uniformly bounded. Hölder gives a common modulus $C\mathrm{\mid }t-s{\mathrm{\mid }}^{1-1\mathrm{/}q}$ (or $C\mathrm{\mid }t-s\mathrm{\mid }$ for $q=\mathrm{\infty }$). Choose a finite time grid on which the modulus between adjacent points is at most $\epsilon$, and finite $\epsilon$-nets for the bounded coefficient vectors at each grid point. The piecewise-linear interpolants of the rounded grid values form a finite family within $2\epsilon$ uniformly of every coefficient vector, so the projected affine family is totally bounded in $C([0,T];E)$. The maximum nodal jump is at most ${\delta }_k^{1-1\mathrm{/}q}\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^q(V^{\ast })}\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_V$, so the projected piecewise-constant family differs from it uniformly by a quantity tending to zero. Thus $\{P_E{\bar{U}}_k\}$ is relatively compact in $L^p(H)$.

The projection remainder satisfies $\mathrm{\parallel }{\bar{U}}_k-P_E{\bar{U}}_k{\mathrm{\parallel }}_{L^p(H)}\le \epsilon \mathrm{\parallel }{\bar{U}}_k{\mathrm{\parallel }}_{L^p(V)}$. Since $\epsilon$ is arbitrary, steps 3.1 and 4.1 show the full family is totally bounded in $L^p(H)$; [A1] extracts a Cauchy subsequence, which converges by [F3]. This proves relative compactness.

If the nodal values are uniformly bounded by $M$ in $H$, then $\mathrm{\parallel }{\bar{U}}_k-{\bar{U}}_{\mathrm{\ell }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(H)}\le 2M$. For $r<p$, finite-measure Hölder upgrades the strong $L^p(H)$ convergence to $L^r(H)$. For $r\ge p$, $\mathrm{\parallel }{\bar{U}}_k-{\bar{U}}_{\mathrm{\ell }}{\mathrm{\parallel }}_{L^r(H)}^r\le (2M{)}^{r-p}\mathrm{\parallel }{\bar{U}}_k-{\bar{U}}_{\mathrm{\ell }}{\mathrm{\parallel }}_{L^p(H)}^p$, so the same subsequence is Cauchy in every finite $L^r(H)$ and converges there by [F3]. These limits equal the strong $L^p(H)$ limit since both convergences imply convergence in measure, whose limit is unique up to almost-everywhere equality.