Scalar finite-element substructuring model

Fix a bounded, connected polygonal open set $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with a nonempty Dirichlet boundary segment. A finite subdomain partition is a family of connected polygonal open sets ${\mathrm{\Omega }}_i$, $1\le i\le N$, whose closures cover $\bar{\mathrm{\Omega }}$, whose interiors are disjoint, and for which ${\bar{\mathrm{\Omega }}}_i\cap {\bar{\mathrm{\Omega }}}_j$ ($i\mathrm{\ne }j$) is a whole common side, a common vertex, or empty. The outer Dirichlet segment is a union of mesh boundary sides. The edge-adjacency graph of each triangulated subdomain is connected, and the subdomain adjacency graph is connected to the Dirichlet boundary. Each subdomain has a matching affine simplicial triangulation; the triangulations match on every shared side. The family has a fixed shape-regularity bound, quasiuniform mesh size $h_i$, uniformly bounded polygon side count, corner angles bounded below, inradius comparable to $H_i$, and a uniform interior-cone parameter. Write $H_i=\mathrm{diam}({\mathrm{\Omega }}_i)$ and $H\mathrm{/}h={\max }_i{H}_i\mathrm{/}{h}_i$.

Use continuous piecewise-affine ($P_1$) scalar functions on each triangulation. The reference-element and face-matching conventions are those of Reference finite elements and unisolvent degrees of freedom, Matching affine simplicial meshes and shape regularity, and Broken and conforming polynomial spaces on a matching mesh: local polynomials are affine, their nodal values determine them, and conforming functions have equal traces across a shared side. For each subdomain choose a coefficient ${\rho }_i>0$, constant on ${\mathrm{\Omega }}_i$ but allowed to jump across interfaces. Its local energy is

$a_i(v,w)={\rho }_i{\sum }_{K\subset {\mathrm{\Omega }}_i}{\int }_K\mathrm{\nabla }v\cdot \mathrm{\nabla }wdx\mathrm{.}$

The global finite-element space consists of the continuous piecewise-affine functions on the subdomain meshes that vanish on the outer Dirichlet boundary, with energy $a(v,w)={\sum }_i{a}_i(v{\mathrm{\mid }}_{{\mathrm{\Omega }}_i},w{\mathrm{\mid }}_{{\mathrm{\Omega }}_i})$. Local interior degrees of freedom are the mesh vertices strictly inside ${\mathrm{\Omega }}_i$; local boundary degrees of freedom are its remaining vertices, with outer Dirichlet values fixed to zero. A floating local block means the same local energy before any outer-boundary values are fixed. A subdomain corner is a vertex of its polygonal boundary.

For uniform estimates, the fixed polygon-shape parameters include the scaled Lipschitz character: after rescaling each subdomain to diameter one, its boundary has a graph-chart cover with uniformly bounded chart count and Lipschitz constants and a uniformly positive neighborhood radius. The scaled mean-zero Poincaré constants are also included among these fixed shape parameters. A pointwise interior-cone bound alone is not a substitute for this uniform boundary-chart data. All constants in later two-dimensional estimates may depend on these polygon-shape, side-count, and mesh shape-regularity bounds, but not on $h$, $H$, the number of subdomains, or the positive numbers ${\rho }_i$.