L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes

The nodal system in [F1] has an invertible interior mass matrix whenever $V_h\mathrm{\ne }\{0\}$; if $V_h=\{0\}$, set $P_{h}f=0$. For finite $r$, every basis function is bounded by a mesh-independent constant and is supported on the cells incident to its node. Hölder and the coefficient scaling in [F1] therefore give, for $b_i=h^{-d}{\int }_{\mathrm{\Omega }}f{\varphi }_i$, $\mathrm{\mid }{b}_i{\mathrm{\mid }}^r\le C{h}^{-d}{\int }_{\mathrm{supp}{\varphi }_i}\mathrm{\mid }f{\mathrm{\mid }}^r,\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{\ell }}^r}\le C{h}^{-d\mathrm{/}r}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^r(\mathrm{\Omega })}\mathrm{.}$ The second inequality follows because on each cell only the fixed number $N(d,k)$ of local basis supports occur, so their overlap is bounded. For $r=\mathrm{\infty }$, $\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$.

For $d=1$, the explicit meshes of $\mathrm{\Omega }=(0,1)$ are $K_j=[j\mathrm{/}N,(j+1)\mathrm{/}N]$, $0\le j<N$. They meet at whole endpoints, $h_N=1\mathrm{/}N$, each inradius is $1\mathrm{/}(2N)$, their shape ratio is $2$, and $\theta =1$; hence $h_N\to 0$. For $d\ge 2$, in every grid cube $a+[0,1\mathrm{/}N{]}^d$, take the $d!$ simplices with cumulative vertices $a,a+N^{-1}{e}_{{\pi }_1},a+N^{-1}(e_{{\pi }_1}+e_{{\pi }_2}),\dots ,a+N^{-1}{\sum }_{j=1}^d{e}_j$ for permutations $\pi$ of $\{1,\dots ,d\}$. In coordinates $t=N(x-a)$ each is $1\ge t_{{\pi }_1}\ge \cdots \ge t_{{\pi }_d}\ge 0$. Sorting coordinates covers the cube and strict orderings have disjoint interiors. For a permutation $\pi$, the barycentric gaps are $g_0=1-t_{{\pi }_1}$, $g_j=t_{{\pi }_j}-t_{{\pi }_{j+1}}$ for $1\le j<d$, and $g_d=t_{{\pi }_d}$; the simplex is $g_j\ge 0$, ${\sum }_j{g}_j=1$. For two permutations, every pair of coordinates ordered oppositely must be equal in their intersection. In either order chain, that equality forces every intervening coordinate equal, hence sets the corresponding consecutive barycentric gaps to zero. The resulting equality blocks have the same relative order in both permutations: otherwise a pair of blocks would be oppositely ordered and would have been merged by an inversion equality. Thus the intersection is exactly the face obtained by setting those gaps to zero, so the within-cube triangulation is matching. On a cube face, coordinates fixed to $1$ occur first and those fixed to $0$ occur last in every compatible ordering; the induced ordering of the remaining coordinates is identical from either adjacent cube. Thus the global simplices meet only in whole faces and form a matching mesh.

If $V_h=\{0\}$, then $P_{h}f=0$ and the $L^r$ bounds are immediate. Otherwise, if $c$ is the coefficient vector of $P_{h}f$, then $A_{h}c=b$. Let $D=A_h^{-1}$, and let $R,C_0$ bound its absolute row and column sums from [F1]. For $1<r<\mathrm{\infty }$, apply [F8] on the finite counting measure to the factors $a_j^{1\mathrm{/}r}\mathrm{\mid }{b}_j\mathrm{\mid }$ and $a_j^{1-1\mathrm{/}r}$, where $a_j=\mathrm{\mid }{D}_{ij}\mathrm{\mid }$. This gives ${\sum }_j{a}_j\mathrm{\mid }{b}_j\mathrm{\mid }\le ({\sum }_j{a}_j\mathrm{\mid }{b}_j{\mathrm{\mid }}^r{)}^{1\mathrm{/}r}({\sum }_j{a}_j{)}^{1-1\mathrm{/}r}$. If the row sum is zero both sides vanish. Raising to $r$ and summing the rows gives ${\sum }_i\mid {\sum }_j{D}_{ij}{b}_j{\mid }^r\le R^{r-1}{\sum }_{i,j}\mathrm{\mid }{D}_{ij}\mathrm{\mid }\mathrm{\mid }{b}_j{\mathrm{\mid }}^r\le R^{r-1}{C}_0{\sum }_j\mathrm{\mid }{b}_j{\mathrm{\mid }}^r\mathrm{.}$ For $r=1$, the triangle inequality and the column sums directly give ${\sum }_i\mathrm{\mid }{\sum }_j{D}_{ij}{b}_j\mathrm{\mid }\le C_0{\sum }_j\mathrm{\mid }{b}_j\mathrm{\mid }$, the same displayed bound. For $r=\mathrm{\infty }$, the row-sum bound gives $\mathrm{\parallel }Db{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}\le R\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}$. Combining these coefficient estimates with [F1] proves the asserted $L^r$ bound for $P_h$. If $f\in L^2$, the mass equations are exactly orthogonality to $V_h$, so $P_h$ is the orthogonal projection and fixes every $w_h\in V_h$.

Each cube simplex contains opposite cube corners and lies in its cube, so its diameter is $h_N=\sqrt{d}\mathrm{/}N$. In row order ${\pi }_1,\dots ,{\pi }_d$, the cumulative-vertex edge matrix has columns $e_{{\pi }_1},e_{{\pi }_1}+e_{{\pi }_2},\dots ,{\sum }_{j=1}^d{e}_{{\pi }_j}$ before scaling by $1\mathrm{/}N$. Subtracting each preceding column from the next gives the permuted identity matrix, so its determinant has absolute value $N^{-d}$; the simplex volume is $1\mathrm{/}(d!N^d)$. Its facets in ordered coordinates are $t_{{\pi }_1}=1$, $t_{{\pi }_d}=0$, and $t_{{\pi }_j}=t_{{\pi }_{j+1}}$. Put $r=1\mathrm{/}(2+(d-1)\sqrt{2})$ and set $t_{{\pi }_j}=r+(d-j)\sqrt{2}r$. The endpoint gaps are $r$ and successive coordinate gaps are $\sqrt{2}r$, so this point is at perpendicular distance $r$ from every facet and is the center of an inscribed ball of radius $r$. For any inscribed ball of radius $s$, its center must have endpoint gaps at least $s$ and successive coordinate gaps at least $\sqrt{2}s$; adding these gaps gives $1\ge 2s+(d-1)\sqrt{2}s$. Thus the inradius is exactly $r\mathrm{/}N$ and the shape ratio is $\sqrt{d}(2+(d-1)\sqrt{2})$, independent of $N$, all cell diameters equal $h_N$ (global quasi-uniformity with $\theta =1$), and only the $d!$ ordered shapes occur. The cube is a bounded Lipschitz polyhedral open set. At a boundary point choose a vector $v$ with $n_j\cdot v>0$ for each active inward face normal $n_j$, and write nearby points as $z+y+sv$ with $y\perp v$. The active inequalities become $s>-n_j\cdot y\mathrm{/}(n_j\cdot v)$; inactive face inequalities remain strict in a sufficiently small neighborhood. Thus locally the cube is the epigraph of the maximum of finitely many affine functions, which is Lipschitz. These are nonempty mesh families satisfying the $d\ge 2$ hypotheses.

Fix $1<p<\mathrm{\infty }$ and $v\in W_0^{1,p}(\mathrm{\Omega })$. For sufficiently fine meshes, since $P_h{Q}_{h}v=Q_{h}v$, the inverse inequality in [F3] and the $L^p$ bound from step 2.1 give $\mathrm{\parallel }\mathrm{\nabla }{P}_{h}v{\mathrm{\parallel }}_p\le \mathrm{\parallel }\mathrm{\nabla }{Q}_{h}v{\mathrm{\parallel }}_p+C{h}^{-1}\mathrm{\parallel }{P}_h(v-Q_{h}v){\mathrm{\parallel }}_p\le \mathrm{\parallel }\mathrm{\nabla }{Q}_{h}v{\mathrm{\parallel }}_p+C{h}^{-1}\mathrm{\parallel }v-Q_{h}v{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p,$ by [F2]. The inverse estimate holds on the interval branch as recorded in [F3]; on simplex meshes summing its cell estimates uses $h_K\ge \theta h$. A sequence with $h\to 0$ has only finitely many coarser initial meshes. On each such mesh, $P_h$ is finite rank and its coefficient functionals $v\mapsto {\int }_{\mathrm{\Omega }}v{\varphi }_i$ are bounded on $L^p$; hence its $W^{1,p}$ output norm is bounded by a mesh-dependent multiple of $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^p}$ by [F8]. Poincaré [F7] bounds this by a multiple of $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p$. Enlarging $C_p$ over the finitely many initial meshes gives the stated uniform estimate.

Step 2.1 and the approximation estimate of [F2] show $\mathrm{\parallel }{P}_{h}v-v{\mathrm{\parallel }}_p\le Ch\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p$. For gradients, write $P_{h}v-v=P_h(v-Q_{h}v)+(Q_{h}v-v)$. Apply the uniform gradient bound proved in the preceding step to $v-Q_{h}v\in W_0^{1,p}$ and the strong approximation in [F2]; both gradient terms tend to zero. Hence $P_{h}v\to v$ in the full $W^{1,p}$ norm.

Let $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Since $Q_h\psi \in V_h$ and $P_h{Q}_h\psi =Q_h\psi$, the $L^2$ bound in step 2.1 gives $\mathrm{\parallel }{P}_h\psi -\psi {\mathrm{\parallel }}_2\le \mathrm{\parallel }{P}_h(\psi -Q_h\psi ){\mathrm{\parallel }}_2+\mathrm{\parallel }{Q}_h\psi -\psi {\mathrm{\parallel }}_2\le (C_2+1)\mathrm{\parallel }{Q}_h\psi -\psi {\mathrm{\parallel }}_{W^{1,2}}⟶0$ by [F2]. For arbitrary $f\in L^2(\mathrm{\Omega })$, choose $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\mathrm{\parallel }f-\psi {\mathrm{\parallel }}_2<\epsilon$ using [F5]. The $L^2$ orthogonal projection is a contraction, so ${\mathrm{lim\hspace{0.17em}sup}}_{h\to 0}\mathrm{\parallel }{P}_{h}f-f{\mathrm{\parallel }}_2\le 2\epsilon \mathrm{.}$ Letting $\epsilon \downarrow 0$ proves strong $L^2$ convergence for every $f\in L^2(\mathrm{\Omega })$.

On each explicit mesh family, $V_h\subset V$: [F4] proves this for matching simplices and the interval membership argument is part of [F2]. The preceding estimates give the $H$-orthogonal projectors, uniform $V$-stability, and strong $V$ approximation; pair the spatial index $N$ with any time partitions whose maximum steps tend to zero. The resulting family satisfies all clauses of [F6], with no coupling between spatial and time refinement.

In the PE12 branch, [A2], [F7], [F9], [F10], and [F11] give the evolution triple with dense continuous compact inclusion. The explicit stable families from step 6.1 therefore instantiate the stated PE12 application for every strict $p$ in the Statement.