LBM-MHFEM chapter - added chapter structure to the introduction

In this work, we investigate a novel computational approach based on the coupling of LBM with the \emph{NumDwarf} scheme described in \cref{chapter:MHFEM}, which is based on the mixed-hybrid finite element method.

As the initial step towards the development of a flexible multi-physics solver, a rather simple model coupling the Navier--Stokes equations with a linear advection--diffusion equation is considered.

The content of this chapter deals with numerical details of the coupled approach based on the paper \cite{klinkovsky2022:WT} and represents original work of the author.

An application of the developed approach to the mathematical modeling of vapor transport in air is described in the next chapter.

%An application of the developed approach to the mathematical modeling of vapor transport in air is described in the next chapter.

An application of the developed approach is described in the next chapter.

The chapter is organized as follows.

\todo{TODO}

Both methods are introduced and their coupling in a time-adaptive manner is explained.

The details related to the decomposition of a regular lattice overlapped with an unstructured mesh are described in \cref{sec:lbm-mhfem:decomposition}.

\Cref{sec:lbm-mhfem:problem formulation} formulates the general problem and its special case with an analytical solution for numerical analysis.

Then, \cref{sec:lbm-mhfem:algorithm,sec:lbm-mhfem:interpolation,sec:lbm-mhfem:decomposition} provide details related to the coupled computational approach and its implementation.

The final \cref{sec:lbm-mhfem:numerical analysis} describes the results of the numerical analysis using the benchmark problem from the first section.

\section{Problem formulation}

\label{sec:lbm-mhfem:problem formulation}

The flow of an incompressible fluid is governed by the Navier--Stokes equations written in the standard form as

The flow of an incompressible fluid is governed by the Navier--Stokes equations

\begin{subequations}\label{eq:lbm-mhfem:ns}

    \begin{equation}\label{eq:lbm-mhfem:ns:mass}

        \nabla \cdot \vec{v} = 0,

    \begin{equation}\label{eq:ADE:conservative}

        \frac{\partial \phi}{\partial t} + \nabla \cdot \left( \phi \vec v - D_0 \nabla \phi \right) = 0, %f_\phi.

    \end{equation}

    where $\phi$ is a variable that may be associated to the physical quantity transported by the fluid (e.g., molar or mass concentration for mass transport, or temperature for heat transport) and $D_0$~[\si{\metre\squared\per\second}] is the corresponding diffusion coefficient.

    where $\phi$ is a variable that may be associated to the physical quantity transported by the fluid (e.g., molar or mass concentration for mass transport, or temperature for heat transport) and $D_0$~[\si{\metre\squared\per\second}] is the diffusion coefficient.

    Combining \cref{eq:ADE:conservative,eq:lbm-mhfem:ns:mass} leads to the non-conservative form

    \begin{equation}\label{eq:ADE:non-conservative}

        \frac{\partial \phi}{\partial t} + \vec v \cdot \nabla \phi - \nabla \cdot \left( D_0 \nabla \phi \right) = 0. %f_\phi,