Stable two-level P1 decomposition for structured Poisson meshes

In a coordinate cube, the stated simplices are the regions where the normalized coordinates have one fixed weak ordering. For each coordinate pair, a coarse ordering hyperplane can meet the interior of a fine cube only when the two fine-cell indices agree; there it is the local equality hyperplane used by the fine Freudenthal triangulation. Thus every fine simplex lies in one coarse simplex. The induced order triangulation on a coordinate face is the same from either incident cube, so the meshes match. The reference simplices come from finitely many coordinate permutations, so their shape-regularity constants are bounded by a number depending only on $d$. By [F3], the continuous P1 spaces are nested. Their boundary nodal values vanish; [F4] places both spaces in $H_0^1(\mathrm{\Omega })$.

Let $J_H$ be the face-average Scott–Zhang map of [F1], set $u_0=J_H{u}_h$, and $w=u_h-u_0\in V_h$. Each coarse element patch ${\omega }_K$ meets only coarse cubes whose coordinate indices differ from that of $K$ by at most one. Thus at most $M_d=3^{d}d!$ coarse simplex patches contain a fixed point. Squaring and summing [F1] over the coarse mesh gives $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C_{S}H\mathrm{\parallel }\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })},\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C_S\mathrm{\parallel }\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ for a constant $C_S$ depending only on $d$ and shape regularity. The second inequality is the global stability part of [F1]. The stated Countable Choice assumption supplies the choice principle required by the Scott–Zhang and Sobolev-trace suppliers in [F1].

For each coarse cube define ${\psi }_j(x)=\max (0,1-\frac{{\mathrm{dist}}_{\mathrm{\infty }}(x,Q_j)}{\delta }),{\eta }_j(x)=\frac{{\psi }_j(x)}{{\sum }_k{\psi }_k(x)}\mathrm{.}$ At every $x\in \mathrm{\Omega }$, at least one coarse cube contains $x$, so the denominator is at least $1$. Since $\delta \le a$, at most $4^d$ functions ${\psi }_k$ are nonzero at a point; each is $1\mathrm{/}\delta$-Lipschitz. The quotient rule for Lipschitz functions with denominator at least $1$ gives $0\le {\eta }_j\le 1$, ${\sum }_j{\eta }_j=1$, and $\mathrm{\parallel }\mathrm{\nabla }{\eta }_j{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le (1+4^d)\mathrm{/}\delta$ almost everywhere. Let ${\theta }_j=I_h{\eta }_j$ be its fine-grid nodal P1 interpolant. The nodal values sum to one, hence ${\sum }_j{\theta }_j=1$ on every fine simplex; barycentric coordinates give $0\le {\theta }_j\le 1$. On a fine simplex with vertices $x_0,\dots ,x_d$ and barycentric coordinates ${\lambda }_i$, $\mathrm{\nabla }{\theta }_j={\sum }_{i=1}^d({\eta }_j(x_i)-{\eta }_j(x_0))\mathrm{\nabla }{\lambda }_i\mathrm{.}$ The Lipschitz bound and [F6] imply $\mathrm{\parallel }\mathrm{\nabla }{\theta }_j{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}\le C_{\theta }\mathrm{/}\delta$ for a constant $C_{\theta }$ depending only on dimension and shape regularity. If ${\theta }_j$ is nonzero on a fine simplex, one of its vertices lies within ${\mathrm{\ell }}^{\mathrm{\infty }}$ distance $\delta$ of $Q_j$, so the whole simplex lies in ${\mathrm{\Omega }}_j$ by the added radius $h$. At most $N=7^d$ expanded coarse cubes contain any point: in each coordinate their side length is at most $a+2(\delta +h)\le 5a$, so at most seven grid intervals can contain that coordinate. Every clipped expanded cube has side lengths between $a$ and $5a$, so its aspect ratio is at most $5$; hence these subdomains are uniformly shape-regular and have diameter at most $5H$.

At every fine-grid vertex, ${\sum }_j{\theta }_{j}w=w$. The nodal interpolant is linear on each fine simplex, so the local functions $u_j=I_h({\theta }_{j}w)$ satisfy ${\sum }_j{u}_j=w$. Each $u_j$ is supported in ${\mathrm{\Omega }}_j$. At a boundary vertex $w=0$, since $u_h,u_0\in V_h$; hence every $u_j$ has zero boundary nodal values and belongs to $H_0^1(\mathrm{\Omega })$ by [F4]. Thus $u_h=u_0+{\sum }_j{u}_j$ is a decomposition in the stated spaces.

Apply [F2] elementwise and $(s+t{)}^2\le 2s^2+2t^2$: ${\sum }_j\mathrm{\parallel }\mathrm{\nabla }{u}_j{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\le 2C_P^{2}N(\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+\frac{C_{\theta }^2}{{\delta }^2}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2)\mathrm{.}$ Indeed, each point belongs to at most $N$ supports, $0\le {\theta }_j\le 1$, and $\mathrm{\mid }\mathrm{\nabla }{\theta }_j\mathrm{\mid }\le C_{\theta }\mathrm{/}\delta$. Also $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2\le (1+C_S)\mathrm{\parallel }\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_2$ by step 1.2.

Substitute the two estimates of step 1.2 into step 2.2 and add $\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_2^2\le C_S^2\mathrm{\parallel }\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_2^2$. With $C_{\mathrm{F}\mathrm{E}}=1+C_S^2+2C_P^{2}N((1+C_S{)}^2+C_{\theta }^2{C}_S^2),$ the result is $\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_2^2+{\sum }_j\mathrm{\parallel }\mathrm{\nabla }{u}_j{\mathrm{\parallel }}_2^2\le C_{\mathrm{F}\mathrm{E}}(1+(H\mathrm{/}\delta {)}^2)\mathrm{\parallel }\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_2^2\mathrm{.}$ This is the asserted stable-decomposition bound.