Coefficient-scaled averaging and BDDC

Use the spaces and partial Schur operator of Primal interface spaces and jump convention and Discrete harmonic extension and interface Schur complement. Let $E_0:\tilde{W}\to \widehat{W}$ average every remaining interface degree of freedom and preserve every already-shared primal value. It is a scaled average if the weights for all copies of each such degree of freedom sum to one. Put $\bar{E}:=R{E}_0:\tilde{W}\to \tilde{W}$. Then $E_{0}R=I_{\widehat{W}}$ and $\bar{E}$ is a projection onto $\widehat{W}$.

In the PE26 two-dimensional corner-constraint branch, each non-primal node is interior to one interface edge shared by exactly two subdomains $i,j$. Set

$(E_{0}w)(x)=\frac{{\rho }_i}{{\rho }_i+{\rho }_j}{w}_i(x)+\frac{{\rho }_j}{{\rho }_i+{\rho }_j}{w}_j(x)\mathrm{.}$

At primal nodes the copies already agree, so the average is the common value. For any finite-dimensional positive definite partial Schur operator $\tilde{S}$, the BDDC preconditioner is

$M_{\mathrm{B}\mathrm{D}\mathrm{D}\mathrm{C}}^{-1}:=E_0{\tilde{S}}^{-1}{E}_0^T:{\widehat{W}}^{\mathrm{\prime }}\to \widehat{W},$

for the assembled interface system with operator $\widehat{S}=R^T\tilde{S}R$. Its constrained partial-energy interpretation is established in BDDC is the constrained partial-energy minimizer ↗.