quad & \text{~if ~} \tilde{U}^{n}_{\alpha,j+\frac{1}{2}} > 0,\\[3ex]
\dfrac{1}{2\mu_{\alpha}}  \left(\rho^{n}_{\alpha,j} + \rho^{n}_{\alpha,j+1}\right) k_{r\alpha}(S^{n}_{\alpha,j+1}),  & \text{~otherwise}.
\end{cases}
\end{equation}
This gives us a non-linear, algebraic system of equations in pressure ($P_{o}$), saturation ($S_{w}$), and Darcy ($U_{\alpha}$) and auxiliary ($\tilde{U}_{\alpha}$) flux unknowns. We now have a system of non-linear algebraic equations that can be linearized using Newton's method (or its variants) to form a linear system of equations that can be solved to obtain the solution. Please note that he flux unknowns will be eliminated using multiple Schur complements of the linearized system of equations resulting in a reduced system in pressure ($P_{o}$) and saturation ($S_{w}$) unknowns only, later in the solution algorithm section. 


\section{Solution Algorithm}\label{sec:algo}
In this section, we present a space-time monolithic solver for the algebraic system obtained in the previous section. We introduce a definition for matching times as the location along the time dimension where the coarse and fine domain boundaries  overlap such that $ \frac{\Delta t_{c}}{\Delta t_{f}} = l \in \mathbb I $, where $l$ is a non-zero integer, $\Delta t_{c}$ the coarse time-step, and $\Delta t_{f}$ the fine time-step. Figure \ref{fig:scheme} shows a schematic of the space-time domain decomposition used for the construction of the monolithic system with $l = 3$. Here, the orange and grey shaded regions represent the fine and coarse, domains respectively both in space and time matching with the description of identifier 1 as discussed in Table 1 before. The schematic shown in Figure \ref{fig:scheme} and the choice of $l=3$ is kept the same in the fully discrete formulation section above to easily describe the relationship between discretization and the resulting linear system sparsity. However, as we will see later in the results section a more general setting can also be treated with more general domain decomposition using a combination of delta change in space and time as well as explicit error estimators as discussed before.
\begin{figure}[H]
\begin{center}
\includegraphics[width=7cm]{st_ev_schematic-eps-converted-to.pdf}\quad
\caption{Space-time domain decomposition schematic and monolithic system construction.}
\label{fig:scheme}
\end{center}
\end{figure}
Now let us turn our attention to the conventional, time-marching solution algorithms for solving PDEs that rely upon previous time-step solution in order to determine the next time-step solution. Such an approach proves restrictive for our space-time adaptive mesh refinement framework with different time-step and grid block sizes in different subdomains of the reservoir. We therefore rely upon a time-concurrent, monolithic solution algorithm that constructs are larger system where fine and coarse domains (in time and irrespective of coarse and fine in space) are solved simultaneously over a prescribed coarse time domain. Let us denote this coarse time-step or more precisely the matching times with $\Delta T$. For the sake of simplicity and to present a coherent argument in line with the fully discrete variational formulation described before, we choose this matching time-step to be the coarse time-step size $\Delta T = \Delta t_{c}$. Again, this choice is not restrictive and a larger matching time-step size ($\Delta T>\Delta t_{c}$) can be chosen without any loss in generality. This monolithic system is then constructed to solve the algebraic equations in the space-time unknowns over the coarse and fine domains for each matching time-step ($\Delta T = \Delta t_{c}$) as follows,
\[
\left[ 
\begin{array}{c@{}c@{}c@{}c@{}c}
 & \vdots & \vdots & \vdots &  \\
   & \textcolor{red}{temporal} & spatial & &  \\
  \hdots & \left[\begin{array}{ccc}
                      \mathcal{T}^{n-\frac{1}{3}}_{pp}&0\\ 
                      0&\mathcal{T}^{n-\frac{1}{3}}_{ss}\\
                      \end{array}\right] 
                      & \left[\begin{array}{ccc}
                      \mathcal{S}^{n-\frac{2}{3}}_{pp}  &  \mathcal{S}^{n-\frac{2}{3}}_{ps} \\ 
                      \mathcal{S}^{n-\frac{2}{3}}_{sp}  & \mathcal{S}^{n-\frac{2}{3}}_{ss} \\
                      \end{array}\right]
                      &\textbf{0}
                      & \hdots \\    
& & \textcolor{red}{temporal} & spatial &   \\     
  \hdots & \textbf{0}
                      & \left[\begin{array}{ccc}
                       \mathcal{T}^{n-\frac{2}{3}}_{pp}&0\\ 
                      0&\mathcal{T}^{n-\frac{2}{3}}_{ss}\\
                      \end{array}\right] 
                      & \left[\begin{array}{ccc}
                      \mathcal{S}^{n}_{pp}  &  \mathcal{S}^{n}_{ps} \\ 
                      \mathcal{S}^{n}_{sp}  & \mathcal{S}^{n}_{ss} \\
                      \end{array}\right]
                      & \hdots \\
                       & \vdots & \vdots & \vdots &
                       \end{array}\right]
\left[
\begin{array}{c@{}}
\vdots\\
\delta x_{i-1}^{n-\frac{1}{3}}\\
\delta x_{i}^{n-\frac{1}{3}}\\
\delta x_{i+1}^{n-\frac{1}{3}}\\
\delta x_{i-1}^{n-\frac{2}{3}}\\
\delta x_{i}^{n-\frac{2}{3}}\\
\delta x_{i+1}^{n-\frac{2}{3}}\\
\delta x_{i-1}^{n}\\
\delta x_{i}^{n}\\
\delta x_{i+1}^{n}\\
\vdots
\end{array}
\right]
=
\left[
\begin{array}{c@{}}
\vdots\\
-\mathcal{R}_{i-1}^{n-\frac{1}{3}}\\
-\mathcal{R}_{i}^{n-\frac{1}{3}}\\
-\mathcal{R}_{i+1}^{n-\frac{1}{3}}\\
-\mathcal{R}_{i-1}^{n-\frac{2}{3}}\\
-\mathcal{R}_{i}^{n-\frac{2}{3}}\\
-\mathcal{R}_{i+1}^{n-\frac{2}{3}}\\
-\mathcal{R}_{i-1}^{n}\\
-\mathcal{R}_{i}^{n}\\
-\mathcal{R}_{i+1}^{n}\\
\vdots
\end{array}
\right].
\] 

Here, $\mathcal{T}$ and $\mathcal{S}$ here represent the temporal and spatial submatrices in the linearized system with subscripts $ps$ and $sp$ indicating pressure-saturation coupling submatrices. Further, $\delta x^{m}_{j} := [\delta P^{m}_{j} \text{ } \delta S_{j}^{m}]^{T}$ is the vector corresponding to the pressure and saturation unknowns and $\mathcal{R}_{j}^{m} := [R_{j,P}^{m} \text{ } R_{j,S}^{m}]$  is the residual vector corresponding to the total and water phase mass conservation equations. Note that here the flux unknowns $\delta U$ and $\delta \tilde{U}$ are eliminated by taking a Schur-complements of the original linear algebraic system in pressure, flux, and saturation unknowns. Also note that the superscripts indicating time vary as $n-1/3$, $n-2/3$, and $n-1$ in agreement with our choice of $\Delta T = \Delta t_{c}$ and the ratio between fine and coarse time-step sizes of $l = \frac{\Delta t_{c}}{\Delta t_{f}} = 3$. This monolithic construction therefore gives us a time-concurrent solution scheme where different time-step sizes in different subdomains of the reservoir can be easily resolved.

Let us now discuss the sparsity of the spatial and temporal matrices in this monolithic system that allows us to relate this scheme to the well known finite difference scheme. For the matching grid case, the spatial sub-matrix here has a known sparsity pattern of three, five, and seven non-zero diagonals for one, two, and three spatial dimensions, respectively similar to the well known finite difference scheme. For the non-matching grid case, the spatial sparsity pattern alters as for the original EVMFEM scheme in space \cite{WheelerEV}. The temporal sub-matrix is always diagonal for our choice of discretization in time. Since this discretization scheme is closely related to the backward