Two-level overlapping P1 Poisson bounds with explicit H-overlap dependence

At almost every point of $\mathrm{\Omega }$, at most $N_0=7^d$ local subdomains are present by [F3] and the coordinate count in Stable two-level P1 decomposition for structured Poisson meshes. For any coarse component $v_0$ and local components $v_j$, at most $N_0+1$ gradients can be nonzero there. [F4] in the finite-dimensional Euclidean inner product gives the pointwise inequality ${\mid \mathrm{\nabla }{v}_0+{\sum }_j\mathrm{\nabla }{v}_j\mid }^2\le (1+N_0)(\mathrm{\mid }\mathrm{\nabla }{v}_0{\mathrm{\mid }}^2+{\sum }_j\mathrm{\mid }\mathrm{\nabla }{v}_j{\mathrm{\mid }}^2)$. Integrating gives ${\parallel \mathrm{\nabla }(v_0+{\sum }_j{v}_j)\parallel }_{L^2(\mathrm{\Omega })}^2\le (1+N_0)(\mathrm{\parallel }\mathrm{\nabla }{v}_0{\mathrm{\parallel }}_2^2+{\sum }_j\mathrm{\parallel }\mathrm{\nabla }{v}_j{\mathrm{\parallel }}_2^2)\mathrm{.}$ Thus $C_1=1+N_0$ is a valid synthesis constant.

Set ${ϵ}_{ij}=1$ when the interiors of ${\mathrm{\Omega }}_i$ and ${\mathrm{\Omega }}_j$ overlap in positive measure and $0$ otherwise, with ${ϵ}_{ii}=1$. If the interiors are disjoint, local gradients have disjoint supports almost everywhere, so $a(v_i,v_j)=0$. If they overlap, [F4] gives $\mathrm{\mid }a(v_i,v_j)\mathrm{\mid }\le \mathrm{\parallel }{v}_i{\mathrm{\parallel }}_A\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_A$. Thus this symmetric nonnegative matrix satisfies the pairwise overlap hypothesis. Since each padded cube has side at most $5a$, two cubes can intersect only when their coarse indices differ by at most five in each coordinate. Thus every row of the overlap matrix has at most $N_1={11}^d$ nonzero entries. For any vector $x$, applying [F4] to the vectors $(\sqrt{{ϵ}_{ij}}{)}_j$ and $(\sqrt{{ϵ}_{ij}}{x}_j{)}_j$ gives $\mathrm{\mid }(ϵx{)}_i{\mathrm{\mid }}^2\le ({\sum }_j{ϵ}_{ij})({\sum }_j{ϵ}_{ij}\mathrm{\mid }{x}_j{\mathrm{\mid }}^2)\le N_1{\sum }_j{ϵ}_{ij}\mathrm{\mid }{x}_j{\mathrm{\mid }}^2$. Summing over $i$ and using the column sums, also at most $N_1$, yields $\mathrm{\parallel }ϵx{\mathrm{\parallel }}_2^2\le N_1^2\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$. Hence $\rho =\mathrm{\parallel }ϵ{\mathrm{\parallel }}_2\le N_1$.

Substituting the stable constant [F1] and $C_1=1+7^d$ from step 1.1 into the additive estimate [F2] proves ${\kappa }_A(BA)\le C_{\mathrm{F}\mathrm{E}}(1+(H\mathrm{/}\delta {)}^2)(1+7^d)$.

Substituting [F1], step 1.1, and $\rho \le {11}^d$ from step 1.2 into the multiplicative estimate [F2] gives the displayed contraction factor. Its denominator is finite and positive, and the multiplicative theorem proves the factor is nonnegative and strictly below one. The bound is uniform as $h\to 0$ and the subdomain count grows whenever $H\mathrm{/}\delta$ stays bounded.