\chapter{Introduction}
\section{Background}
In 1856, the French engineer Henry Darcy formulated the Darcy's law for flow in a porous medium based on his experiments of water flowing through beds of sand.
\begin{figure}[htbp]
\begin{center}
\resizebox{4in}{1.2in}{\includegraphics{Darcyfig1.png}}
\caption{Darcy's law}\label{Darcyfiglaw}
\end{center}
\end{figure}
In Figure (\ref{Darcyfiglaw}), $Q$ $(\mbox{m}^3/\mbox{s})$ is the total discharge, $A $ $(\mbox{m}^2)$ is the cross-sectional area to flow, $(p_b-p_a)$ (pascals) is the total pressure drop, $\nabla l$ (m) is the length over the pressure drop. Darcy's law was introduced by Darcy \cite{BookSIAM_Ewing_1983} and then refined by Morris Muskat. It becomes
\begin{equation}
Q = -\frac{\kappa A(p_b-p_a)}{\mu \nabla l},
\end{equation}
where $\kappa $ $(\mbox{m}^2)$ is the permeability of the medium, $\mu$ $(\mbox{Pa}\cdot \mbox{s})$ is the viscosity of fliud, the negative sign comes from the reason that the fluid flows from higher pressure to lower pressure.
Darcy's law has become an important tool in analysis of the ground water flow and is widely used in the areas of hydrodynamics, oil recovery, chemical engineering and many other engineering fields. The Darcy equation coupled with the elasticity equation has been used to describe flow in a poroelastic medium.
%the poroelasticity equation, interaction of solid and fluid equation, which could be solved by finite element methods.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Outline}
This thesis concentrates on a typical boundary value problem defined by the Darcy equation on a domain $ \Omega $. The Darcy equation is formulated as
\begin{equation}
\label{EqnDarcy}
\left\{
\begin{array}{l}
\displaystyle
\nabla \cdot \left( -\mathbf{K} \nabla p \right)
\equiv \nabla \cdot \mathbf{u} = f,
\quad \mathbf{x} \in \Omega,
\\ [0.05in]
\displaystyle
p = p_D,
\quad \mathbf{x} \in \Gamma^D,
\qquad
\mathbf{u} \cdot \mathbf{n} = u_N,
\quad \mathbf{x} \in \Gamma^N,
\end{array}
\right.
\end{equation}
where $\Omega \subset \mathbb{R}^n (n=2,3)$ is a bounded domain. In the context of the flow of a fluid through a porous medium, $p$ is the pressure, $\mathbf{K}$ is a permeability tensor, $\mathbf{u} = -\mathbf{K} \nabla p$ is the Darcy velocity which is the flow per unit cross sectional area of the porous medium, $ f $ is the source term,
$ p_D, u_N $ are respectively Dirichlet and Neumann boundary data.
Several methods have been developed for solving the Darcy equation, such as continuous Galerkin finite element methods (CGFEMs) and mixed finite element methods (MFEMs) \cite{BastRivie_IJNMF_2003, Chen_iFEM_2009, BookSIAM_Ewing_1983,WihlerRivie_JSC_2011}. For both these two methods, we ultimately need to solve a large-scale linear system. Although CGFEMs have fewer unknowns, they are also known to lack local mass conservation and flux continuity. MFEMs solve the unknown pressure and the velocity simultaneously, but they need to satisfy the inf-sup condition and result in an indefinite linear system.
Here, we use the weak Galerkin finite element methods (WGFEMs) introduced in {\cite{WangYe_JCAM_2013} }to solve the Darcy equation.
This thesis is organized as follows. In Chapter 2, we present the existing finite element methods, CGFEMs and MFEMs, for solving the Darcy equation. In Chapter 3, WGFEMs are presented in detail, such as the construction of weak Galerkin finite element schemes on quadrilateral and triangular meshes and numerical experiments. Chapter 4 discusses the WG method for three-dimensional flows. Chapter 5 describes the future work for WGFEMs and Chapter 6 concludes this thesis.
%\smallskip
%These are the references used by this thesis,
%\cite{GinLinLiu_JSC_2016} %1
%
%\cite{LinLiuMuYe_JCP_2014} %2
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%\cite{LiuMuYe_PCS_2011} %3
%
%\cite{AhnCarey_IJNMF_2007} %4
%
%\cite{AlberCarsFunken_NumerAlgor_1999} %5
%
%\cite{ArboCorrea_SINUM_2016} %6
%
%\cite{ArnoldBoffiFalk_MCOM_2002} %7
%
%\cite{ArnoldBoffiFalk_SINUM_2005} %8
%
%\cite{BahriaCars_CMAM_2005} %9
%
%\cite{BastRivie_IJNMF_2003} %10
%
%\cite{BookSpringer_BrezziFortin_1991} %11
%
%\cite{BushGin_SISC_2013} %12
%
%\cite{Chen_iFEM_2009} %13
%
%\cite{ChouHe_CMAME_2002} %14
%
%\cite{CockGopaWang_SINUM_2007} %15
%
%\cite{Durlof_WRR_1994} %16
%
%\cite{EwingLiuWang_SINUM_1999} %17
%
%\cite{BookSIAM_Ewing_1983} %18
%
%\cite{GinLinLiu_JSC_2016} %19
%
%\cite{HarperLiuZheng_ICCS_2017} %20
%
%\cite{LinLiuSadre_JCAM_2015}%21
%
%\cite{LiuSadreWang_ICCS_2016}%22
%
%\cite{MuWangYe_JCAM_2015}%23
%
%\cite{MuWangYe_IJNAM_2015}%24
%
%\cite{SunLiu_SISC_2009}%25
%
%\cite{WangYe_JCAM_2013}%26
%
%\cite{WheeXueYotov_NumerMath_2012}%27
%
%\cite{WheeYotov_SINUM_2006}%28
%
%\cite{WihlerRivie_JSC_2011}%29