Finite-dimensional resolved–unresolved VMS split and residual closure

Let $X_c$ and $X_f$ be finite-dimensional real vector spaces and set $X=X_c\oplus X_f$; elements are written $(u_c,u_f)$, with $u_c$ the resolved component and $u_f$ the unresolved component. For a real vector space $Y$, let $Y^{\ast }$ be its space of real linear functionals. Let $A_{cc}:X_c\to X_c^{\ast }$, $A_{cf}:X_f\to X_c^{\ast }$, $A_{fc}:X_c\to X_f^{\ast }$, and $A_{ff}:X_f\to X_f^{\ast }$ be linear maps, with $A_{ff}$ invertible, and let $F_c\in X_c^{\ast }$ and $F_f\in X_f^{\ast }$. The linear equation on $X$ has block form $\left(\begin{array}{cc}{A}_{cc}& A_{cf}\\ A_{fc}& A_{ff}\end{array}\right)\left(\begin{array}{c}{u}_c\\ u_f\end{array}\right)=\left(\begin{array}{c}{F}_c\\ F_f\end{array}\right)\mathrm{.}$ The fine residual for a resolved vector $v_c$ is $r_f(v_c):=F_f-A_{fc}{v}_c$. Exact elimination gives $u_f=A_{ff}^{-1}{r}_f(u_c)$ and the Schur equation $(A_{cc}-A_{cf}{A}_{ff}^{-1}{A}_{fc})u_c=F_c-A_{cf}{A}_{ff}^{-1}{F}_f\mathrm{.}$

A residual closure selects a linear map $T:X_f^{\ast }\to X_f$ to approximate the exact inverse $A_{ff}^{-1}:X_f^{\ast }\to X_f$. It defines ${\tilde{A}}_{cc}:=A_{cc}-A_{cf}T{A}_{fc}$, ${\tilde{F}}_c:=F_c-A_{cf}T{F}_f$, and the modeled fine component ${\tilde{u}}_f:=T{r}_f({\tilde{u}}_c)$, where ${\tilde{u}}_c$ solves ${\tilde{A}}_{cc}{\tilde{u}}_c={\tilde{F}}_c$. This is a finite-dimensional algebraic closure; an application to a PDE requires its own residual, spaces, invertibility, and approximation-defect analysis.