Energy and computed-state compactness for hybrid mimetic backward Euler

The quantity $B_{h,p}$ is a norm on the finite zero-boundary cell-face space: if it vanishes, $G_K=0$ and ${\rho }_{K,F}=0$, so $v_K=v_F$; connectedness and the boundary faces force every value to zero. Coercivity in [F2] and the lower stabilization bound give ${\sum }_K{\mathcal{A}}_{h,K}(v;v)\ge c{B}_{h,p}(v{)}^p-\mathrm{\parallel }{h}_0{\mathrm{\parallel }}_1$, writing $h_0$ for the flux's coercivity datum. At a fixed step the mass term paired with $v$ is bounded below by $-C_h\mathrm{\parallel }{\mathrm{\Pi }}_h{U}^{n-1}{\mathrm{\parallel }}_2{B}_{h,p}(v)$, and the source term is at most $C\mathrm{\parallel }{f}^n{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}{B}_{h,p}(v)$ by [F1]. The averaged cell flux is continuous in its finite-dimensional arguments by its growth bound and dominated convergence on bounded argument sets. Thus the equation's coordinate residual is continuous and its pairing with $v$ is positive on every sufficiently large Euclidean sphere, since finite-dimensional norm equivalence and $p>1$ make the $B_{h,p}^p$ term dominate the linear terms. Brouwer [F2] gives a solution, inductively at every step. No mesh-uniform sphere radius is needed for existence.

Test each step by $U^n$ and use $2(a-b)a=a^2-b^2+(a-b{)}^2$ in the volume-weighted cell inner product. Hölder and [F1] bound the source by $C\mathrm{\parallel }{f}^n{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}{B}_{h,p}(U^n)$, which is at most $(c\mathrm{/}2)B_{h,p}(U^n{)}^p+C\mathrm{\parallel }{f}^n{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}^{p^{\mathrm{\prime }}}$: this scalar Young bound follows by maximizing $xy-(c\mathrm{/}2)y^p$ over $y\ge 0$. Multiply by ${\tau }_n$ and sum to any terminal index. Jensen for the cell/time averages gives ${\sum }_n{\tau }_n\mathrm{\parallel }{f}^n{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}^{p^{\mathrm{\prime }}}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(Q)}^{p^{\mathrm{\prime }}}$. Coercivity contributes at worst $T\mathrm{\parallel }{h}_0{\mathrm{\parallel }}_1$. The initial norms are bounded by strong initialization, giving all three displayed energy bounds and the $L^p$ bound on $B_{h,p}$.

For $z\in V$, test a time step by $I_{h}z$. Because its cell values are the exact cell means, the mass term is exactly ${\int }_{\mathrm{\Omega }}{D}_t({\mathrm{\Pi }}_h{U}^n)z$ and the load is exactly ${\int }_{\mathrm{\Omega }}{f}^{n}z$. Growth and Hölder give $\mathrm{\mid }{\sum }_K{\mathcal{A}}_{h,K}(U^n;I_{h}z)\mathrm{\mid }\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}+B_{h,p}(U^n{)}^{p-1})\mathrm{\parallel }z{\mathrm{\parallel }}_V$: the flux term uses $\mathrm{\parallel }{G}_h{I}_{h}z{\mathrm{\parallel }}_p\le C\mathrm{\parallel }z{\mathrm{\parallel }}_V$, and the stabilization term uses its weighted Hölder bound $C{S}_{h,p}(U^n{)}^{p-1}{S}_{h,p}(I_{h}z)$. Hence $\mathrm{\parallel }{D}_t{\mathrm{\Pi }}_h{U}^n{\mathrm{\parallel }}_{V^{\ast }}\le C(\mathrm{\parallel }{f}^n{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}+\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}+B_{h,p}(U^n{)}^{p-1})$. Since $(p-1)p^{\mathrm{\prime }}=p$, step 2.1 gives the asserted derivative bound for the affine cell-state interpolation.

Fix $\epsilon >0$. Compactness $V⋐H$ gives a finite-dimensional subspace $Z\subset V$ and its $H$-orthogonal projection $P_Z$ such that $\mathrm{\parallel }(I-P_Z)v{\mathrm{\parallel }}_H\le \epsilon \mathrm{\parallel }v{\mathrm{\parallel }}_V$ for all $v\in V$: take a finite $\epsilon$-net of the $V$ unit ball's image and span its centers. By [F1], $\mathrm{\parallel }(I-P_Z){\mathrm{\Pi }}_h{\bar{U}}_h{\mathrm{\parallel }}_{L^p(H)}\le C(h^{\vartheta }+\epsilon )$. Each coefficient $({\mathrm{\Pi }}_h{U}^n,z{)}_H$, $z$ in a fixed orthonormal basis of $Z$, is bounded uniformly in $n$ by step 2.1, and its affine interpolant has derivative bounded in $L^{p^{\mathrm{\prime }}}$ by step 3.1. Hölder gives its common modulus $C\mathrm{\mid }t-s{\mathrm{\mid }}^{1\mathrm{/}p}$. Finite-grid rounding therefore makes these finitely many coefficients totally bounded in the uniform norm. Their largest nodal jumps are at most $C{\delta }_h^{1\mathrm{/}p}$, so the piecewise constant coefficients have the same compactness. The projection remainder tends to zero as first $h\to 0$ and then $\epsilon \downarrow 0$. Completeness of $L^p(H)$ and AC extract strong convergence there. This proves compactness for the actual cell states without falsely applying a conforming-state theorem to them.

The companions differ from the cell states by $C{h}^{\vartheta }$ in $L^p(H)$ and are bounded in $L^p(V)$. Thus [F3] gives $u\in L^p(V)$ and weak companion convergence. The uniform cell-state $L^{\mathrm{\infty }}(H)$ bound passes to $u$ by almost-everywhere $H$ convergence along a further subsequence. It upgrades strong $L^p(H)$ convergence to every finite $L^r(H)$: finite-measure Hölder treats $r<p$, while $\mathrm{\parallel }{v}_h-u{\mathrm{\parallel }}_{L^r(H)}^r\le (2M{)}^{r-p}\mathrm{\parallel }{v}_h-u{\mathrm{\parallel }}_{L^p(H)}^p$ treats $r\ge p$. The bounded fields $G_h{\bar{U}}_h$ have weak $L^p$ subsequences; the weak-gradient defect in [F1], integrated in time, identifies every such limit as $\mathrm{\nabla }u$.

Compactness $V⋐L^p$ and continuous $V\to H$ give for each $\epsilon >0$ an Ehrling estimate $\mathrm{\parallel }z{\mathrm{\parallel }}_p\le \epsilon \mathrm{\parallel }z{\mathrm{\parallel }}_V+C_{\epsilon }\mathrm{\parallel }z{\mathrm{\parallel }}_H$. Indeed, if this failed, normalize counterexamples to $\mathrm{\parallel }{z}_j{\mathrm{\parallel }}_p=1$, with $\mathrm{\parallel }{z}_j{\mathrm{\parallel }}_V\le 1\mathrm{/}\epsilon$ and $\mathrm{\parallel }{z}_j{\mathrm{\parallel }}_H\to 0$. Compactness in $L^p$ gives a limit of norm one, while the $H$ convergence makes that limit zero as a distribution, a contradiction. Apply this estimate to $E_h{\bar{U}}_h-u$, whose $L^p(V)$ norm is bounded and whose $L^p(H)$ norm tends to zero. It tends to zero in $L^p(Q)$ after $h\to 0$ and then $\epsilon \downarrow 0$. Its $L^p(Q)$ distance from ${\mathrm{\Pi }}_h{\bar{U}}_h$ is $O(h)$ by [F1]. This proves the remaining strong convergence. All arguments used only $h\to 0$ and ${\delta }_h\to 0$ separately.