FETI-DP dual system and preconditioner

Use $\tilde{W}$, $\widehat{W}$, $E_0$, $R$, $B$ and $\mathrm{\Lambda }=\mathrm{range}B$ as defined in Primal interface spaces and jump convention and Coefficient-scaled averaging and BDDC. Assume that $\tilde{S}:\tilde{W}\to {\tilde{W}}^{\mathrm{\prime }}$ is symmetric positive definite. In the PE26 corner branch this follows from Primal constraints remove local floating kernels when all subdomain vertices are primal and the global Dirichlet boundary is nonempty; for other primal choices it is a separate admissibility requirement. Use the inherited Euclidean inner product to identify $\mathrm{\Lambda }$ with ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ in the transpose and coordinate formulas below. Regard $\bar{E}=R{E}_0$ as a projection on $\tilde{W}$. Since $\ker B=\mathrm{range}\bar{E}=\widehat{W}$, each class of traces with the same jump has a unique representative in $\ker \bar{E}$. Define $B_D^T:\mathrm{\Lambda }\to \tilde{W}$ by assigning to $\lambda \in \mathrm{\Lambda }$ that unique representative $w$ with $Bw=\lambda$ and $\bar{E}w=0$. This is the scaled right inverse; it obeys $B{B}_D^T=I_{\mathrm{\Lambda }}$ and $B_D^{T}B=I_{\tilde{W}}-\bar{E}$. The latter identity makes the primal and dual projections complementary. In the two-subdomain edge case, its two entries are the coefficient-scaled jump weights complementary to those in $E_0$.

The FETI-DP dual operator and its preconditioner are

$F:=B{\tilde{S}}^{-1}{B}^T:\mathrm{\Lambda }\to {\mathrm{\Lambda }}^{\mathrm{\prime }},M_{\mathrm{F}\mathrm{E}\mathrm{T}\mathrm{I}-\mathrm{D}\mathrm{P}}^{-1}:=B_D\tilde{S}{B}_D^T:{\mathrm{\Lambda }}^{\mathrm{\prime }}\to \mathrm{\Lambda }\mathrm{.}$

The multiplier convention removes dependent row coordinates: $\lambda$ is in $\mathrm{range}B$, so no arbitrary multiplier component in $\ker B^T$ is retained. Elimination of the partially assembled displacement and the resulting exact dual system are proved in FETI-DP is the exact dual Schur system ↗.