Finite-dimensional backward-Euler diffusion transmission model

Let $I_1,I_2$ and the common trace space $\mathrm{\Lambda }$ be finite-dimensional real inner-product spaces, with $\dim \mathrm{\Lambda }\ge 1$, and set $X_i:=I_i\oplus \mathrm{\Lambda }$. The trace map is ${\gamma }_i(x_i,{\lambda }_i)={\lambda }_i$. Let $M_i$ and $A_i$ be self-adjoint positive-definite operators on $X_i$, let $\mathrm{\Delta }t>0$, and define the local backward-Euler block $B_i:=\mathrm{\Delta }{t}^{-1}{M}_i+A_i$. In coordinates $I_i\oplus \mathrm{\Lambda }$, write

$B_i=\left(\begin{array}{cc}{B}_{i,00}& B_{i,0\mathrm{\Gamma }}\\ B_{i,\mathrm{\Gamma }0}& B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}\end{array}\right),g_i=(g_{i,0},g_{i,\mathrm{\Gamma }})\mathrm{.}$

For a prescribed local trace ${\lambda }_i$, solve the interior equation $B_{i,00}{x}_i+B_{i,0\mathrm{\Gamma }}{\lambda }_i=g_{i,0}$ and define the outward algebraic interface flux $q_i:=B_{i,\mathrm{\Gamma }0}{x}_i+B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}{\lambda }_i-g_{i,\mathrm{\Gamma }}$. Eliminating $x_i$ gives $q_i=S_i{\lambda }_i-f_i$, where $S_i:=B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}-B_{i,\mathrm{\Gamma }0}{B}_{i,00}^{-1}{B}_{i,0\mathrm{\Gamma }}$ and $f_i:=g_{i,\mathrm{\Gamma }}-B_{i,\mathrm{\Gamma }0}{B}_{i,00}^{-1}{g}_{i,0}$. The sign convention is outward flux on each subdomain.

The monolithic interface equations are trace continuity ${\lambda }_1={\lambda }_2=\lambda$ and flux balance $q_1+q_2=0$, equivalently $(S_1+S_2)\lambda =f_1+f_2$. These finite-dimensional matrices represent a linear diffusion-transmission step; no nonlinear fluid-structure well-posedness is asserted.