Conditional monotone finite-element convergence

The orthogonal projections are contractions on $H$, and for any $v\in V$, ${\mathrm{\Pi }}_{k}v\to v$ in $V$ and hence in $H$. Given $u_0\in H$, choose $v\in V$ close to $u_0$ using density in [F1]. Then $\mathrm{\parallel }{\mathrm{\Pi }}_k{u}_0-u_0{\mathrm{\parallel }}_H\le 2\mathrm{\parallel }{u}_0-v{\mathrm{\parallel }}_H+\mathrm{\parallel }{\mathrm{\Pi }}_{k}v-v{\mathrm{\parallel }}_H\to 2\mathrm{\parallel }{u}_0-v{\mathrm{\parallel }}_H\mathrm{.}$ Letting $v$ approach $u_0$ proves ${\mathrm{\Pi }}_k{u}_0\to u_0$ in $H$. Also [F4] gives $f_k\to f$ strongly in $L^{p^{\mathrm{\prime }}}(V^{\ast })$ for the piecewise-constant averages $f_k(t)=f_k^n$ on $I_k^n$.

By [F3], the reconstructions ${\bar{U}}_k$ are uniformly bounded in $L^p(V)$ and $L^{\mathrm{\infty }}(H)$, and their discrete derivatives $D_k$ are uniformly bounded in $L^{p^{\mathrm{\prime }}}(V^{\ast })$. The maximum partition step tends to zero, so [F5] gives a subsequence converging strongly in $L^p(H)$ and, by the $L^{\mathrm{\infty }}(H)$ bound, in every finite $L^r(H)$. Applying [F6] to this subsequence gives a further subsequence and $u\in L^p(V)$ such that ${\bar{U}}_k\rightharpoonup u$ in $L^p(V)$, its image in $L^p(H)$ is the strong limit, and $\mathrm{\nabla }{\bar{U}}_k\rightharpoonup \mathrm{\nabla }u$ in $L^p(Q;{\mathbb{R}}^d)$.

If $u$ and $\tilde{u}$ are two solutions with the same data, their difference $w=u-\tilde{u}$ has zero initial value and satisfies $w^{\mathrm{\prime }}+A(u)-A(\tilde{u})=0$. By [F8], for every $t$, $\frac{1}{2}\mathrm{\parallel }w(t){\mathrm{\parallel }}_H^2+{\int }_0^t⟨A(u)-A(\tilde{u}),w⟩=0.$ Pointwise monotonicity in [F1] makes the integral nonnegative, so $w(t)=0$ for every $t$. This proves uniqueness in the abstract conclusion.

Put $q_k(t,x)=a(x,\mathrm{\nabla }{\bar{U}}_k(t,x))$. The gradients are measurable, so [F14] and the growth hypothesis [F1] give $\mathrm{\parallel }{q}_k{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(Q)}\le C(T^{1\mathrm{/}{p}^{\mathrm{\prime }}}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })}+\mathrm{\parallel }\mathrm{\nabla }{\bar{U}}_k{\mathrm{\parallel }}_{L^p(Q)}^{p-1}),$ so $(q_k)$ is bounded in vector-valued $L^{p^{\mathrm{\prime }}}(Q)$. Each of its $d$ coordinates is bounded in scalar $L^{p^{\mathrm{\prime }}}(Q)$ and hence has a weakly convergent subsequence by [F7]. Successive extraction over the finite coordinate set gives one subsequence weakly convergent in every coordinate; restriction of any bounded functional on the finite product to its coordinate embeddings shows weak convergence of the vector fields. Write the limit as $q\in L^{p^{\mathrm{\prime }}}(Q;{\mathbb{R}}^d)$.

For each fixed $v\in V$ and $\varphi \in C_c^{\mathrm{\infty }}(0,T)$, integrate the discrete equation against $\varphi$ and test with ${\mathrm{\Pi }}_{k}v$. If ${\widehat{U}}_k$ is the piecewise-affine reconstruction, integration by parts gives $-{\int }_0^T({\widehat{U}}_k,{\mathrm{\Pi }}_{k}v{)}_H{\varphi }^{\mathrm{\prime }}+{\int }_Q{q}_k\cdot \mathrm{\nabla }{\mathrm{\Pi }}_{k}v\varphi ={\int }_0^T⟨f_k,{\mathrm{\Pi }}_{k}v⟩\varphi \mathrm{.}$ On each cell, $\mathrm{\parallel }{\widehat{U}}_k-{\bar{U}}_k{\mathrm{\parallel }}_{V^{\ast }}\le {\tau }_k^n\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{V^{\ast }}$, hence $\mathrm{\parallel }{\widehat{U}}_k-{\bar{U}}_k{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })}\le {\delta }_k\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })}\to 0.$ Now [F2] gives ${\mathrm{\Pi }}_{k}v\to v$ in $V$; steps 1.2 and 2.1 give the strong state convergence and weak flux convergence; [F4] gives strong convergence of $f_k$. Hölder [F11] therefore permits passage in every term, yielding $-{\int }_0^T(u,v{)}_H{\varphi }^{\mathrm{\prime }}+{\int }_{Q}q\cdot \mathrm{\nabla }v\varphi ={\int }_0^T⟨f,v⟩\varphi \mathrm{.}$ Thus the distributional derivative of $u$ is $f-B_q$, where $⟨B_q(t),v⟩={\int }_{\mathrm{\Omega }}q(t,x)\cdot \mathrm{\nabla }v(x)dx$ and $B_q\in L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })$.

The uniform bound $\mathrm{\parallel }{\bar{U}}_k(t){\mathrm{\parallel }}_H\le M$ and strong $L^p(H)$ convergence imply $(\mathrm{\parallel }u(t){\mathrm{\parallel }}_H-M{)}_{+}\le \mathrm{\parallel }u(t)-{\bar{U}}_k(t){\mathrm{\parallel }}_H$ almost everywhere. Taking $L^p$ norms and passing to the limit shows $u\in L^{\mathrm{\infty }}(0,T;H)$. Step 3.1 places $u$ in $L^p(V)$ with derivative $f-B_q\in L^{p^{\mathrm{\prime }}}(V^{\ast })$; [F8] therefore gives its continuous $H$ representative and energy identity.

Fix $v\in V$. The scalar affine interpolants ${\alpha }_k(t)=({\widehat{U}}_k(t),v{)}_H$ are uniformly bounded, and ${\alpha }_k^{\mathrm{\prime }}(t)=⟨D_k(t),v⟩$ is uniformly bounded in $L^{p^{\mathrm{\prime }}}$. Hölder gives a common modulus $\mathrm{\mid }{\alpha }_k(t)-{\alpha }_k(s)\mathrm{\mid }\le C_v\mathrm{\mid }t-s{\mathrm{\mid }}^{1\mathrm{/}p}$. The reconstruction defect in step 3.1 and the strong state convergence imply ${\alpha }_k\to \alpha :=(u,v{)}_H$ in $L^1(0,T)$; $\alpha$ is continuous by step 4.1. This convergence is uniform: if $\mathrm{\mid }{\alpha }_k(t_k)-\alpha (t_k)\mathrm{\mid }\ge \epsilon$ along a subsequence, the common modulus and continuity of $\alpha$ give a fixed $\delta >0$ on which the difference is at least $\epsilon \mathrm{/}3$ around $t_k$. The intersected interval has length at least $\min (\delta ,T)$, contradicting $L^1$ convergence. Therefore $(U_k^{N_k},v{)}_H={\alpha }_k(T)\to (u(T),v{)}_H$.

Use a smooth $\varphi$ with $\varphi (0)=1$ and $\varphi (T)=0$. Integrating the discrete equation by parts in time gives $-(U_k^0,{\mathrm{\Pi }}_{k}v{)}_H-{\int }_0^T({\widehat{U}}_k,{\mathrm{\Pi }}_{k}v{)}_H{\varphi }^{\mathrm{\prime }}+{\int }_Q{q}_k\cdot \mathrm{\nabla }{\mathrm{\Pi }}_{k}v\varphi ={\int }_0^T⟨f_k,{\mathrm{\Pi }}_{k}v⟩\varphi \mathrm{.}$ The convergence arguments of step 3.1 and ${\mathrm{\Pi }}_k{u}_0\to u_0$ from step 1.1 pass this identity to the limit. By step 3.1 the distributional derivative of $t\mapsto (u(t),v{)}_H$ is $⟨f-B_q,v⟩\in L^1(0,T)$; [F12] gives its absolutely continuous representative, which agrees with the continuous $H$ representative from step 4.1. Its integration-by-parts formula gives the same limit identity with $u(0)$ in place of $u_0$. Therefore $(u(0),v{)}_H=(u_0,v{)}_H$ for every $v\in V$, and density of $V$ in [F1] proves $u(0)=u_0$.

The nodal values are uniformly bounded in $H$ by [F3]. Step 5.1 gives $U_k^{N_k}\rightharpoonup u(T)$ against every $v\in V$; density of $V$ in $H$ and the uniform $H$ bounds extend this to weak convergence in $H$. Consequently $\mathrm{\parallel }u(T){\mathrm{\parallel }}_H^2\le {\mathrm{lim\hspace{0.17em}inf}}_k\mathrm{\parallel }{U}_k^{N_k}{\mathrm{\parallel }}_H^2$: if $u(T)\mathrm{\ne }0$, test the weak convergence against $u(T)$ and apply Cauchy–Schwarz; the zero case is immediate.

Testing each discrete equation with $U_k^n$, summing the exact identity [F10], and dropping the nonnegative jump sum gives ${\int }_Q{q}_k\cdot \mathrm{\nabla }{\bar{U}}_k\le {\int }_0^T⟨f_k,{\bar{U}}_k⟩+\frac{1}{2}\mathrm{\parallel }{U}_k^0{\mathrm{\parallel }}_H^2-\frac{1}{2}\mathrm{\parallel }{U}_k^{N_k}{\mathrm{\parallel }}_H^2\mathrm{.}$ Strong convergence $f_k\to f$ and weak convergence ${\bar{U}}_k\rightharpoonup u$ make the first term on the right converge to ${\int }_0^T⟨f,u⟩$. Steps 1.1 and 6.1 therefore imply ${\mathrm{lim\hspace{0.17em}sup}}_k{\int }_Q{q}_k\cdot \mathrm{\nabla }{\bar{U}}_k\le {\int }_0^T⟨f,u⟩+\frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_H^2-\frac{1}{2}\mathrm{\parallel }u(T){\mathrm{\parallel }}_H^2\mathrm{.}$ The continuous equation from step 3.1 and the energy identity in step 4.1 make the right-hand side equal to ${\int }_{Q}q\cdot \mathrm{\nabla }u$.

Apply [F9] to the weak gradient convergence in step 1.2, the weak flux convergence in step 2.1, and the limsup inequality in step 7.1. It gives $q=a(x,\mathrm{\nabla }u)$ almost everywhere. Hence $B_q=A(u)$ and step 3.1 is the claimed weak evolution equation.

For the unit interval in $d=1$, [F13] supplies the projector sequence for every $p>1$, which is the stated strict threshold. For the unit cube in any $d\ge 2$, [F13] supplies it for every $p>\max (1,2d\mathrm{/}(d+2))$. Choose any time partitions with maximum step tending to zero, independently of the spatial index. The resulting mesh/projector/time sequences satisfy every hypothesis used in the preceding convergence proof, so the abstract convergence and uniqueness conclusion holds on these explicit nonempty standard-mesh families without an additional projector assumption.