The two-dimensional BDDC and FETI-DP polylogarithmic bound

Apply [F2] with $L=E_0$ and $T=R$. The BDDC eigenvalues are at least one and at most $\mathrm{\parallel }R{E}_0{\mathrm{\parallel }}_{\tilde{S}}^2=\mathrm{\parallel }I-P{\mathrm{\parallel }}_{\tilde{S}}^2$. By the triangle inequality for the induced operator norm, $\mathrm{\parallel }I-P{\mathrm{\parallel }}_{\tilde{S}}\le 1+\mathrm{\parallel }P{\mathrm{\parallel }}_{\tilde{S}}\le 1+C_0(1+\log (H\mathrm{/}h))$. Since $1+\log (H\mathrm{/}h)\ge 1$, squaring yields the BDDC bound with a geometry-only constant.

Apply [F2] with $L=B$ and $T=B_D^T$. This bounds the eigenvalues of $F{M}_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}$ below by one and above by $\mathrm{\parallel }{B}_D^{T}B{\mathrm{\parallel }}_{\tilde{S}}^2=\mathrm{\parallel }P{\mathrm{\parallel }}_{\tilde{S}}^2\le C_0^2(1+\log (H\mathrm{/}h){)}^2$ by [F1] and [F4]. The Gram factors $F=B{\tilde{S}}^{-1}{B}^T$ and $M_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}=B_D\tilde{S}{B}_D^T$ are positive definite: $B^T$ is injective on $\mathrm{\Lambda }$ by [F4], $B_D^T$ is injective by $B{B}_D^T=I_{\mathrm{\Lambda }}$, and $\tilde{S}$ is positive definite. Thus both maps are isomorphisms, and the stated product $M_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}F$ has the same spectrum because $F(M_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}F)F^{-1}=F{M}_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}\mathrm{.}$

The lower eigenvalue bounds bound both condition numbers by their respective largest eigenvalues. The common nonunit spectrum in [F5] confirms that the paired methods have the same nontrivial convergence spectrum; neither assertion introduces a 3D, arbitrary-graph or unaligned-coefficient claim excluded in the Statement.