LBM-MHFEM chapter - updated references to the numerical scheme in the MHFEM chapter

The impact of using the linear or cubic interpolation on the accuracy of the numerical solution is investigated in \cref{sec:lbm-mhfem:numerical analysis}.

The finite element space used by MHFEM imposes requirements on the interpolation of the velocity field.

According to \cite{brezzi:1991mixed}, the Raviart--Thomas--Nédélec space of the lowest order $\mathbf{RTN}_0(\mathcal K_h)$ that is used for the finite element--approximation of the velocity field in this work is formed by functions $\vec \omega \in [L^2(\Omega_2)]^3$ such that:

In this work, we use the Raviart--Thomas--Nédélec space of the lowest order $\mathbf{RTN}_0(\mathcal K_h)$ for the finite element--approximation of the velocity field.

According to the definition in \cref{eq:RTN0:global_definition} on \cpageref{eq:RTN0:global_definition}, functions $\vec \omega$ belonging to the space $\mathbf{RTN}_0(\mathcal K_h)$ must satisfy:

\begin{enumerate}

    \item for any element $K \in \mathcal K_h$, the restriction of $\vec \omega$ to $K$, $\vec \omega|_K$, must belong to the finite element space $\mathbf{RTN}_0(K)$ on the element $K$,

    \item

        $\vec \omega$ satisfies the balancing condition for normal traces on all interior faces $E \in \mathcal E_h^{\mathrm{int}}$ of the mesh, i.e.,

        for any element $K \in \mathcal K_h$, the restriction of $\vec \omega$ to $K$, $\vec \omega|_K$, must belong to the finite element space $\mathbf{RTN}_0(K)$ on the element $K$,

    \item

        the normal trace of $\vec \omega$ must be continuous on all interior faces of the mesh, i.e.,

        $\int_E \vec \omega|_{K_1} \cdot \vec n_{K_1,E} + \int_E \vec \omega|_{K_2} \cdot \vec n_{K_2,E} = 0$

        for all $E \in \mathcal E_h^{\mathrm{int}}$, $E \in \mathcal E_{K_1} \cap \mathcal E_{K_2}$, where $\vec n_{K_\ell,E}$ is the unit normal vector on the face $E$ oriented outward from the element $K_\ell$, $\ell=1,2$.

\end{enumerate}

In this section, we study numerically the convergence of the coupled LBM-MHFEM scheme using an artificial benchmark problem described in \cref{sec:lbm-mhfem:problem formulation}.

The aim of this section is to study the differences between the conservative and non-conservative formulations of the advection--diffusion equation \eqref{eq:ADE}.

Several variants of the MHFEM scheme from \cref{chapter:MHFEM} were used, namely explicit or implicit upwind, and linear or cubic interpolation of the velocity field.

Several variants of the MHFEM scheme from \cref{chapter:MHFEM} were used, namely explicit or implicit upwind (see \cref{sec:mhfem:upwind}), and linear or cubic interpolation of the velocity field (see \cref{sec:lbm-mhfem:interpolation}).

Each variant was computed in three resolutions denoted as RES-A1, RES-A2, and RES-A3, see \cref{tab:ADE2:resolutions}.

Note that single precision was used in the LBM part for fluid flow and double precision was used in the MHFEM part.

To illustrate the turbulent flow field in $\Omega_1$, \cref{fig:ADE2:vx} shows the horizontal velocity ($v_x$) field in the final time $t_{\max} = \SI{10}{\second}$.

Both qualitative and quantitative results in \cref{fig:ADE2:concentrations,tab:ADE2:norms} indicate that for the conservative formulation, changing linear interpolation to cubic, as well as changing the explicit upwind discretization to implicit upwind, leads to smoother and more accurate results.

Furthermore, all these variants converge to the analytical solution as the lattice and grid are refined.

However, even the most accurate numerical solution obtained using the conservative formulation exhibits an error that is larger by orders of magnitude compared to the non-conservative formulation, even in the coarsest resolution.

The only difference between the discretizations of the non-conservative and conservative formulations is in \cref{eq:mhfem:advection terms discrete}\todo{fix reference when the term is added to \cref{sec:mhfem:numerical scheme}} where the former contains a term corresponding to the discrete divergence of velocity.

The only difference between the discretizations of the non-conservative and conservative formulations is in \cref{eq:mhfem:advection terms discrete} on \cpageref{eq:mhfem:advection terms discrete} where the former contains a term corresponding to the discrete divergence of velocity.

The results indicate that this extra term can be understood as a compensation for the non-zero divergence of the discrete velocity field interpolated on the mesh.

Furthermore, it can be noticed in \cref{tab:ADE2:norms} that changing the interpolation or upwind scheme does not have a significant effect on the error when the non-conservative formulation is used.

In the finest resolution RES-A3, using the linear interpolation and explicit upwind is not only advantageous for the performance of the solver, but also leads to a smaller error.

\end{align}

\end{subequations}

To modify the terms containing vector coefficients, we use \cref{eq:mhfem:vectors_approximation}, the properties of the basis functions in \cref{eq:RTN0:basis_conditions,eq:dg:scalar_basis_properties}, and the Green's formula:

\begin{subequations}

\begin{subequations} \label{eq:mhfem:advection terms discrete}

\begin{align}

    \int\limits_\Omega \vec u_{i,j} \cdot \nabla Z_j \varphi_K

    &= \int\limits_{\partial K} Z_j \varphi_K \vec u_{i,j} \cdot \vec n_{\partial K}