Nonlinear backward Euler Galerkin existence and energy bounds

Fix $k$. If $V_k=\{0\}$, then $U_k^0={\mathrm{\Pi }}_k{u}_0=0$, and setting $U_k^n=0$ for every $n$ satisfies every equation (the only test vector is zero). Assume $V_k\mathrm{\ne }\{0\}$. For a time cell $I_k^n$, the Bochner average $f_k^n\in V^{\ast }$ exists. Define $F_n:V_k\to V_k^{\ast }$ by $⟨F_n(z),w_k⟩={\tau }_k^{-1}(z-U_k^{n-1},w_k{)}_H+⟨A(z)-f_k^n,w_k⟩\mathrm{.}$ Choose a fixed $H$-orthonormal basis ${\psi }_1,\dots ,{\psi }_N$ of $V_k$ and identify $V_k^{\ast }$ with ${\mathbb{R}}^N$ by its values on that basis. To prove continuity, let $z_m={\sum }_j{c}_{mj}{\psi }_j$ with $c_{mj}\to c_j$, and put $z={\sum }_j{c}_j{\psi }_j$. The coefficients are bounded by some $M$, so $\mathrm{\mid }\mathrm{\nabla }{z}_m\mathrm{\mid }\le M{\sum }_j\mathrm{\mid }\mathrm{\nabla }{\psi }_j\mathrm{\mid }=:B\in L^p(\mathrm{\Omega })$ and the gradients converge pointwise outside one null set. For each test basis vector ${\psi }_i$, Carathéodory continuity gives pointwise convergence of $a(x,\mathrm{\nabla }{z}_m)\cdot \mathrm{\nabla }{\psi }_i$, while growth bounds its absolute value by $C(g+B^{p-1})\mathrm{\mid }\mathrm{\nabla }{\psi }_i\mathrm{\mid }\in L^1$ by Hölder [F5]. Thus [F7] proves convergence of every flux test coordinate. The mass and load coordinates are continuous and affine, so $F_n$ is a continuous finite-dimensional map, without a global operator-continuity assumption. Finite dimensionality gives constants $c_k,C_k>0$ with $c_k\mathrm{\parallel }z{\mathrm{\parallel }}_H\le \mathrm{\parallel }z{\mathrm{\parallel }}_V\le C_k\mathrm{\parallel }z{\mathrm{\parallel }}_H$ on $V_k$. Coercivity [F1] therefore gives, with $r=\mathrm{\parallel }z{\mathrm{\parallel }}_H$, $⟨F_n(z),z⟩\ge {\tau }_k^{-1}{r}^2+\alpha c_k^p{r}^p-({\tau }_k^{-1}\mathrm{\parallel }{U}_k^{n-1}{\mathrm{\parallel }}_H+C_k\mathrm{\parallel }{f}_k^n{\mathrm{\parallel }}_{V^{\ast }})r-\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1}\mathrm{.}$ Since $p>1$, the right side is positive for all sufficiently large $r$.

Apply [F3] on such a sphere in the chosen coordinates to obtain $F_n(U_k^n)=0$. Starting with $U_k^0={\mathrm{\Pi }}_k{u}_0$, repeat this finite step for $n=1,\dots ,N_k$; this constructs the discrete solution on the partition.

Test the equation with $v_k=U_k^n$. By [F4], $\frac{1}{2}(\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_H^2-\mathrm{\parallel }{U}_k^{n-1}{\mathrm{\parallel }}_H^2+\mathrm{\parallel }{U}_k^n-U_k^{n-1}{\mathrm{\parallel }}_H^2)+{\tau }_k^n\alpha \mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_V^p\le {\tau }_k^n\mathrm{\parallel }{f}_k^n{\mathrm{\parallel }}_{V^{\ast }}\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_V+{\tau }_k^n\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1}\mathrm{.}$ Use [F5] to absorb half the coercive term and sum over $n$. Hölder on each cell gives ${\sum }_n{\tau }_k^n\mathrm{\parallel }{f}_k^n{\mathrm{\parallel }}_{V^{\ast }}^{p^{\mathrm{\prime }}}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(V^{\ast })}^{p^{\mathrm{\prime }}}$. Consequently $\frac{1}{2}\mathrm{\parallel }{U}_k^m{\mathrm{\parallel }}_H^2+\frac{1}{2}{\sum }_{n\le m}\mathrm{\parallel }{U}_k^n-U_k^{n-1}{\mathrm{\parallel }}_H^2+\frac{\alpha }{2}{\sum }_{n\le m}{\tau }_k^n\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_V^p\le \frac{1}{2}\mathrm{\parallel }{U}_k^0{\mathrm{\parallel }}_H^2+T\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1}+C_{\alpha ,p}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(V^{\ast })}^{p^{\mathrm{\prime }}},$ for every $m$. Orthogonality gives $\mathrm{\parallel }{U}_k^0{\mathrm{\parallel }}_H\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_H$; taking the maximum over $m$ bounds the first three quantities uniformly in $k$.

For $v\in V$, orthogonality and $D_k{\mathrm{\mid }}_{I_k^n}\in V_k$ give $⟨D_k,v⟩=(D_k,iv{)}_H=(D_k,{\mathrm{\Pi }}_{k}v{)}_H=⟨f_k^n-A(U_k^n),{\mathrm{\Pi }}_{k}v⟩\mathrm{.}$ Thus [F2] yields $\mathrm{\parallel }{D}_k(t){\mathrm{\parallel }}_{V^{\ast }}\le C_{\mathrm{\Pi }}(\mathrm{\parallel }{f}_k^n{\mathrm{\parallel }}_{V^{\ast }}+\mathrm{\parallel }A(U_k^n){\mathrm{\parallel }}_{V^{\ast }})$. The flux growth bound [F1] and the inequality $(a+b{)}^{p^{\mathrm{\prime }}}\le 2^{p^{\mathrm{\prime }}-1}(a^{p^{\mathrm{\prime }}}+b^{p^{\mathrm{\prime }}})$ give $\mathrm{\parallel }{D}_k{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(V^{\ast })}^{p^{\mathrm{\prime }}}\le C_{p,C_1}{C}_{\mathrm{\Pi }}^{p^{\mathrm{\prime }}}(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(V^{\ast })}^{p^{\mathrm{\prime }}}+T\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })}^{p^{\mathrm{\prime }}}+{\sum }_n{\tau }_k^n\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_V^p)\mathrm{.}$ Here $(p-1)p^{\mathrm{\prime }}=p$, and $g$ is constant in time. Step 3.1 bounds the last sum, proving the derivative estimate with precisely the stated family and data dependencies.

For the fixed-degree interval and cube sequences in the Statement, [F6] verifies the abstract FEM projector hypotheses, including the independent time partitions. The preceding construction and energy estimates therefore apply to these concrete nonempty meshes.