diff --git a/doc/handbook/0_dumux-handbook.tex b/doc/handbook/0_dumux-handbook.tex
index 99e3a3faedaf1b70e79358766293df1a11923c3d..7d8185ce932364fe073dc33f783f213f183c70f1 100644
--- a/doc/handbook/0_dumux-handbook.tex
+++ b/doc/handbook/0_dumux-handbook.tex
@@ -73,6 +73,7 @@
\DeclareMathOperator{\grad}{\mathbf{grad}}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\Div}{div}
+\newcommand{\meas}[1]{\lvert{#1}\rvert}
\pagestyle{scrheadings}
diff --git a/doc/handbook/5_models.tex b/doc/handbook/5_models.tex
index 6da2bedb66e2bf88b332d0e85686f01a162dcdc1..6fc5e5319374193c59173003d57a21056bcbfd78 100644
--- a/doc/handbook/5_models.tex
+++ b/doc/handbook/5_models.tex
@@ -261,5 +261,5 @@ In comparison to solving the equations monolothically, the sequential structure
different discretization methods for the different equations. The standard method
used in the sequential algorithm is a cell-centered finite volume method. Further schemes,
so far only available for the two-phase pressure equation, are cell-centered finite
-volumes with multi-point flux approximation (MPFA O-method) and mimetic finite differences.
+volumes with multi-point flux approximation (Mpfa-O method) and mimetic finite differences.
An $h$-adaptive implementation of both sequential algorithms is provided for two dimensions.
diff --git a/doc/handbook/5_spatialdiscretizations.tex b/doc/handbook/5_spatialdiscretizations.tex
index 77b93ac2eab82f6aa3e7b4080af983d1fe95f2ca..d45be682e004bb7f22ca4fba56425d8fa09ecd16 100644
--- a/doc/handbook/5_spatialdiscretizations.tex
+++ b/doc/handbook/5_spatialdiscretizations.tex
@@ -45,13 +45,13 @@ In the following, the discretization of the balance equation is going to be deri
From the \textsc{Reynolds} transport theorem follows the general balance equation:
\begin{equation}
- \underbrace{\int_\Omega \frac{\partial}{\partial t} \: u \: dx}_{1}
- + \underbrace{\int_{\partial\Omega} (\mathbf{v} u + \mathbf w) \cdot \textbf n \: d\varGamma}_{2} = \underbrace{\int_\Omega q \: dx}_{3}
+ \underbrace{\int_\Omega \frac{\partial}{\partial t} \: u \, \mathrm{d}x}_{1}
+ + \underbrace{\int_{\partial\Omega} (\mathbf{v} u + \mathbf w) \cdot \textbf n \, \mathrm{d}\Gamma}_{2} = \underbrace{\int_\Omega q \, \mathrm{d}x}_{3}
\end{equation}
\begin{equation}
- f(u) = \int_\Omega \frac{\partial u}{\partial t} \: dx + \int_{\Omega} \nabla \cdot
- \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \: dx - \int_\Omega q \: dx = 0
+ f(u) = \int_\Omega \frac{\partial u}{\partial t} \, \mathrm{d}x + \int_{\Omega} \nabla \cdot
+ \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \, \mathrm{d}x - \int_\Omega q \, \mathrm{d}x = 0
\end{equation}
where term 1 describes the changes of entity $u$ within a control volume over
time, term 2 the advective, diffusive and dispersive fluxes over the interfaces
@@ -67,30 +67,30 @@ this means:
\begin{minipage}[b]{0.47\textwidth}
\begin{equation}
\label{eq:p}
- \tilde p = \sum_i N_i \hat{p_i}
+ \tilde p = \sum_i N_i \hat{p}_i
\end{equation}
\begin{equation}
\label{eq:v}
- \tilde{\mathbf v} = \sum_i N_i \hat{\mathbf v}
+ \tilde{\mathbf v} = \sum_i N_i \hat{\mathbf v}_i
\end{equation}
\begin{equation}
\label{eq:x}
- \tilde x^\kappa = \sum_i N_i \hat x^\kappa
+ \tilde x^\kappa = \sum_i N_i \hat x_i^\kappa
\end{equation}
\end{minipage}
\hfill
\begin{minipage}[b]{0.47\textwidth}
\begin{equation}
\label{eq:dp}
- \nabla \tilde p = \sum_i \nabla N_i \hat{p_i}
+ \nabla \tilde p = \sum_i \nabla N_i \hat{p}_i
\end{equation}
\begin{equation}
\label{eq:dv}
- \nabla \tilde{\mathbf v} = \sum_i \nabla N_i \hat{\mathbf v}
+ \nabla \tilde{\mathbf v} = \sum_i \nabla N_i \hat{\mathbf v}_i
\end{equation}
\begin{equation}
\label{eq:dx}
- \nabla \tilde x^\kappa = \sum_i \nabla N_i \hat x^\kappa .
+ \nabla \tilde x^\kappa = \sum_i \nabla N_i \hat x_i^\kappa .
\end{equation}
\end{minipage}
@@ -106,103 +106,231 @@ of the residual $\varepsilon$ with a weighting function $W_j$ and claiming that
this product has to vanish within the whole domain,
\begin{equation}
- \int_\Omega W_j \cdot \varepsilon \: \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1
+ \int_\Omega \varepsilon W_j \, \mathrm{d}x \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1
\end{equation}
yields the following equation:
\begin{equation}
- \int_\Omega W_j \frac{\partial \tilde u}{\partial t} \: dx + \int_\Omega W_j
- \cdot \left[ \nabla \cdot F(\tilde u) \right] \: dx - \int_\Omega W_j
- \cdot q \: dx = \int_\Omega W_j \cdot \varepsilon \: dx \: \overset {!}{=} \: 0 .
+ \int_\Omega \frac{\partial \tilde u}{\partial t} W_j \, \mathrm{d}x + \int_\Omega
+ \left[ \nabla \cdot F(\tilde u) \right] W_j \, \mathrm{d}x - \int_\Omega q W_j \, \mathrm{d}x = \int_\Omega \varepsilon W_j \, \mathrm{d}x \: \overset {!}{=} \: 0.
+\label{eq:weightedResidual}
\end{equation}
-Then, the chain rule and the \textsc{Green-Gaussian} integral theorem are applied.
+For standard Galerkin schemes, the weighting functions $W_j$ are chosen the same as the ansatz functions $N_j$. However, this does not yield a locally mass-conservative scheme.
+Therefore, for the Box method, the weighting functions $W_j$ are chosen as
+the piecewise constant functions over a
+control volume box $B_j$, i.e.
\begin{equation}
- \int_\Omega W_j \frac{\partial \sum_i N_i \hat u_i}{\partial t} \: dx
- + \int_{\partial\Omega} \left[ W_j \cdot F(\tilde u)\right]
- \cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega \nabla W_j \cdot F(\tilde u)
- \: dx - \int_\Omega W_j \cdot q \: dx = 0
+ W_j(x) = \begin{cases}
+ 1 &x \in B_j \\
+ 0 &x \notin B_j.\\
+ \end{cases}
+\label{eq:weightingFunctions}
\end{equation}
+Thus, the Box method is a Petrov-Galerkin scheme, where the weighting functions do not belong to the same function space than the ansatz functions.
-A mass lumping technique is applied by assuming that the storage capacity is
-reduced to the nodes. This means that the integrals $M_{i,j} = \int_\Omega W_j \: N_i \: dx$
-are replaced by the mass lumping term $M^{lump}_{i,j}$ which is defined as:
+Inserting definition \eqref{eq:weightingFunctions} into equation \eqref{eq:weightedResidual} and using the \textsc{Green-Gaussian} integral theorem results in
+\begin{equation}
+ \int_{B_j} \frac{\partial \tilde u}{\partial t} \, \mathrm{d}x + \int_{\partial B_j} F(\tilde u) \cdot \mathbf n \, \mathrm{d}\Gamma_{B_j} - \int_{B_j} q \, \mathrm{d}x \overset {!}{=} \: 0,
+\label{eq:BoxMassBlance}
+\end{equation}
+which has to hold for every box $B_j$.
+The first term in equation \eqref{eq:BoxMassBlance} can be written as
\begin{equation}
- M^{lump}_{i,j} =\begin{cases} \int_\Omega W_j \: dx = \int_\Omega N_i \: dx = V_i &i = j\\
- 0 &i \neq j\\
+\int_{B_j} \frac{\partial \tilde u}{\partial t} \, \mathrm{d}x = \frac{d}{dt} \int_{B_j} \sum_i \hat u_i N_i \, \mathrm{d}x = \sum_i \frac{\partial \hat u_i}{\partial t} \int_{B_j} N_i \, \mathrm{d}x.
+\end{equation}
+Here, a mass lumping technique is applied by assuming that the storage capacity is
+reduced to the nodes. This means that the integrals $M_{i,j} = \int_{B_j} N_i \, \mathrm{d}x$
+are replaced by some mass lumped terms $M^{lump}_{i,j}$ which are defined as
+\begin{equation}
+ M^{lump}_{i,j} =\begin{cases} |B_j| &j = i\\
+ 0 &j \neq i,\\
\end{cases}
\end{equation}
-where $V_i$ is the volume of the FV box $B_i$ associated with node $i$.
-The application of this assumption in combination with
-$\int_\Omega W_j \:q \: dx = V_i \: q$ yields
+where $|B_j|$ is the volume of the FV box $B_j$ associated with node $j$.
+The application of this assumption yields
\begin{equation}
- V_i \frac{\partial \hat u_i}{\partial t}
- + \int_{\partial\Omega} \left[ W_j \cdot F(\tilde u)\right]
- \cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega \nabla W_j \cdot F(\tilde u)
- \: dx- V_i \cdot q = 0 \, .
+\label{eq:disc1}
+ |B_j| \frac{\partial \hat u_j}{\partial t}
+ + \int_{\partial B_j} F(\tilde u) \cdot \mathbf n \, \mathrm{d}\Gamma_{B_j} - Q_j = 0,
\end{equation}
+where $Q_j$ is an approximation (using some quadrature rule) of the integrated source/sink term $\int_{B_j} q \, \mathrm{d}x$.
-Defining the weighting function $W_j$ to be piecewisely constant over a
-control volume box $B_i$
+Using an implicit Euler time discretization finally
+leads to the discretized form which will be applied to the mathematical
+flow and transport equations:
\begin{equation}
- W_j(x) = \begin{cases}
- 1 &x \in B_i \\
- 0 &x \notin B_i\\
- \end{cases}
+\label{eq:discfin}
+ |B_j| \frac{\hat u_j^{n+1} - \hat u_j^{n}}{\Delta t}
+ + \int_{\partial B_j} F(\tilde u^{n+1}) \cdot \mathbf n
+ \; \mathrm{d}\Gamma_{B_j} - Q_j^{n+1} \: = 0.
\end{equation}
+Equation \eqref{eq:discfin} has to be fulfilled for each box $B_j$.
-causes $\nabla W_j = 0$:
+\subsection{Cell Centered Finite Volume Methods -- A Short Introduction}\label{cc}
+Cell-centered finite volume methods use the elements of the grid as control volumes.
+For each control volume the discrete values are determined at the element/control
+volume center (not required to be the barycenters).
+We consider a domain $\Omega \subset \mathbb{R}^d$, $d \in \{ 2, 3 \}$ with boundary $\Gamma = \partial \Omega$. Within this section, we consider the following elliptic problem
\begin{equation}
-\label{eq:disc1}
- V_i \frac{\partial \hat u_i}{\partial t}
- + \int_{\partial B_i} \left[ W_j \cdot F(\tilde u)\right]
- \cdot \mathbf n \; d{\varGamma}_{B_i} - V_i \cdot q = 0 .
+ \begin{aligned}
+ \nabla \cdot \left( - \mathbf{\Lambda} \nabla u \right) &= q &&\mathrm{in} \, \Omega \\
+ \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} &= v_N &&\mathrm{on} \, \Gamma_N \\
+ u &= u_D &&\mathrm{on} \, \Gamma_D.
+ \label{eq:elliptic}
+ \end{aligned}
\end{equation}
-The consideration of the time discretization and inserting $W_j = 1$ finally
-leads to the discretized form which will be applied to the mathematical
-flow and transport equations:
+Here, $\mathbf{\Lambda} = \mathbf{\Lambda}(\mathbf{x}, \mathbf{u})$ is a symmetric and positive definite tensor of second rank (e.g. permeability, diffusivity, etc.), $u = u (\mathbf{x})$ is unknown and $q = q(\mathbf{x}, \mathbf{u})$ is a source/sink.
+We denote by $\mathcal{M}$ the mesh that results from the division of the domain $\Omega$ into $n_e$ control volumes $K \subset \Omega$. Each $K$ is a polygonal open set such that $K \cap L = \emptyset, \forall{K \neq L}$ and $\overline{\Omega} = \cup_{K \in \mathcal{M}} \overline{K}$.
+
+For the derivation of the finite-volume formulation we integrate the first equation of \eqref{eq:elliptic} over a control volume $K$ and apply the Gauss divergence theorem:
\begin{equation}
-\label{eq:discfin}
- V_i \frac{\hat u_i^{n+1} - \hat u_i^{n}}{\Delta t}
- + \int_{\partial B_i} F(\tilde u^{n+1}) \cdot \mathbf n
- \; d{\varGamma}_{B_i} - V_i \: q^{n+1} \: = 0
+ \int_{\partial K} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \, \mathrm{d} \Gamma = \int_K q \, \mathrm{d}x.
+ \label{eq:ellipticIntegrated}
\end{equation}
-\subsection{Cell Centered Finite Volume Method -- A Short Introduction}\label{cc}
+Splitting the control volume boundary $\partial K$ into a finite number of faces $\sigma \subset \partial K$ (such that $\sigma = \overline{K} \cap \overline{L}$ for some neighboring control volume $L$) and replacing the exact fluxes by an approximation, i.e. $F_{K, \sigma} \approx \int_{\sigma} \left( - \mathbf{\Lambda}_K \nabla u \right) \cdot \mathbf{n} \mathrm{d} \Gamma$ (here $\mathbf{\Lambda}_K$ is the value of $\mathbf{\Lambda}$ associated with control volume $K$), yield
+\begin{equation}
+ \sum_{\sigma \subset \partial K} F_{K, \sigma} = Q_K, \quad \forall \, {K \in \mathcal{M}},
+\label{eq:ccdisc}
+\end{equation}
+where $F_{K, \sigma}$ is the discrete flux through face $\sigma$ flowing out of cell $K$ and $Q_K := \int_K q \, \mathrm{d}x$ is the integrated source/sink term. Equation \eqref{eq:ccdisc} is the typical cell-centered finite-volume formulation.
+Finite-volume schemes differ in the way how the term
+$(\mathbf{\Lambda}_K \nabla u ) \cdot \mathbf{n} $ is approximated (i.e. the choice of the fluxes $F_{K, \sigma}$). Using the symmetry of the tensor $\mathbf{\Lambda}_K$, this term can be rewritten as
+$\nabla u \cdot \mathbf{\Lambda}_K\mathbf{n}$, which corresponds to the directional derivative of $u$ in co-normal direction $\mathbf{\Lambda}_K\mathbf{n}$.
+In the following, the main ideas of the two-point flux approximation and the multi-point flux approximation methods are briefly described. Hereby, we restrict the discussion to the two-dimensional case.
+
+Please also note that other types of equations, e.g. instationary parabolic problems, can be discretized by applying some time discretization scheme to the time derivatives and by using the finite-volume scheme for the flux discretization. For simplicity the discussion is restricted to the elliptic problem \eqref{eq:elliptic}.
+
+\subsubsection{Tpfa Method}\label{cc_tpfa}
+The linear two-point flux approximation is a simple but robust cell-centered finite-volume scheme, which is commonly used in commercial software.
+This scheme can be derived by using the conormal decomposition, which reads
+\begin{equation}
+\mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} = t_{K,\sigma} \mathbf{d}_{K,\sigma} + \mathbf{d}^{\bot}_{K,\sigma}, \quad t_{K,\sigma} = \frac{\mathbf{n}_{K, \sigma}^T \mathbf{\Lambda}_K \mathbf{d}_{K,\sigma} }{\mathbf{d}_{K,\sigma}^T \mathbf{d}_{K,\sigma}}, \; \mathbf{d}^{\bot}_{K,\sigma} = \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} - t_{K,\sigma} \mathbf{d}_{K,\sigma},
+\label{eq:conormalDecTpfa}
+\end{equation}
+with the tensor $\mathbf{\Lambda}_K$ associated with control volume $K$, the distance vector $\mathbf{d}_{K,\sigma} := \mathbf{x}_\sigma - \mathbf{x}_K$ and $\mathbf{d}_{K,\sigma}^T \mathbf{d}^{\bot}_{K,\sigma} = 0$, see Figure \ref{pc:cctpfa} for the used notations. The same can be done for the conormal $\mathbf{\Lambda}_L \mathbf{n}_{L, \sigma}$. The $t_{K,\sigma}$ and $t_{L,\sigma}$ are the transmissibilities associated with the face $\sigma$. These transmissibilities are calculated in \Dumux by using the function \texttt{computeTpfaTransmissibility}.
\begin{figure} [ht]
\centering
-\includegraphics[width=0.4\linewidth,keepaspectratio]{png/cc_disc.png}
-\caption{\label{pc:cc} Discretization of the cell centered finite volume method}
+\includegraphics[width=0.4\linewidth,keepaspectratio]{png/cctpfa.png}
+\caption{Two neighboring control volumes sharing the face $\sigma$.}
+\label{pc:cctpfa}
\end{figure}
-The cell centered finite volume method uses the elements of the grid as control volumes.
-For each control volume all discrete values are determined at the element/control
-volume center (see Figure~\ref{pc:cc}).
-The mass or energy fluxes are evaluated at the integration points ($x_{ij}$),
-which are located at the midpoints of the control
-volume faces. This is a two point flux approximation since the flux between
-the element/control volume centers $i$ and $j$ is calculated
-only with information from these two points. In contrast the box method uses
-a multi-point flux approximation where all nodes of the
-element influence the flux between two specific nodes. \par
-Neumann boundary conditions are applied at the boundary control volume faces
-and Dirichlet boundary conditions at the boundary control volumes. \par
-The cell centered finite volume method is robust and mass conservative but
-should only be applied for structured grids
-(the control volume face normal vector ($n_{ij}$) should be parallel to the
-direction of the gradient between the two element/control
-volume centers).
-
-% \subsubsection{MPFA}\label{cc_mpfa}
-% TODO
+
+With these notations, it follows that for each cell $K$ and face $\sigma$
+\begin{equation}
+\nabla u \cdot \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} = t_{K,\sigma} \nabla u \cdot \mathbf{d}_{K,\sigma} + \nabla u \cdot \mathbf{d}^{\bot}_{K,\sigma}.
+\end{equation}
+For the Tpfa scheme, the second part in the above equation is neglected. By using the fact that $\nabla u \cdot \mathbf{d}_{K,\sigma} \approx u_\sigma - u_K$, the discrete fluxes for face $\sigma$ are given by
+\begin{equation}
+F_{K,\sigma} = -\meas{\sigma} t_{K,\sigma} (u_\sigma - u_K), \qquad F_{L,\sigma} = -\meas{\sigma} t_{L,\sigma} (u_\sigma - u_L).
+\label{eq:TPFAOneSided}
+\end{equation}
+Enforcing local flux conservation, i.e. $F_{K,\sigma}+F_{L,\sigma}=0$, results in
+\begin{equation}
+u_\sigma = \frac{t_{K,\sigma} u_K + t_{L,\sigma} u_L}{t_{K,\sigma} + t_{L,\sigma}}.
+\end{equation}
+With this, the fluxes \eqref{eq:TPFAOneSided} are rewritten as
+\begin{equation}
+F_{K,\sigma} = \meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} (u_K - u_L), \quad F_{L,\sigma} = \meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} (u_L - u_K).
+\label{eq:TPFAFlux}
+\end{equation}
+By neglecting the orthogonal term, the consistency of the scheme is lost for general grids, where $\nabla u \cdot \mathbf{d}^{\bot}_{K,\sigma} \not = 0$. The consistency is achieved only for so-called K-orthogonal grids for which $\mathbf{d}^{\bot}_{K,\sigma} = 0$. For such grids we deduce that
+\begin{equation}
+\frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} = \frac{\tau_{K,\sigma} \tau_{L,\sigma}}{\tau_{K,\sigma} d_{L,\sigma} + \tau_{L,\sigma} d_{K,\sigma}},
+\label{eq:TPFAcoeffNew}
+\end{equation}
+with $\tau_{K,\sigma} := \mathbf{n}_{K, \sigma} \mathbf{\Lambda}_K\mathbf{n}_{K, \sigma}, \tau_{L,\sigma} := \mathbf{n}_{L, \sigma} \mathbf{\Lambda}_L\mathbf{n}_{L, \sigma}$, $d_{K,\sigma}:= \mathbf{n}_{K, \sigma} \cdot \mathbf{d}_{K, \sigma}$, and $d_{L,\sigma}:= \mathbf{n}_{L, \sigma} \cdot \mathbf{d}_{L, \sigma}$. This reduces, for the case of scalar permeability, to a distance weighted harmonic averaging of permeabilities.
+
+
+
+\subsubsection{Mpfa Method}\label{cc_mpfa}
+Expressions for the face fluxes $F_{K, \sigma}$ are obtained by introducing intermediate face unknowns $u_\sigma$ in addition to the cell unknowns $u_K$ and enforcing the physically motivated continuity of fluxes and continuity of the solution across the faces. For a face $\sigma$ between the two polygons $K$ and $L$ these conditions read:
+\begin{equation}
+ \begin{aligned}
+ &F_{K, \sigma} + F_{L, \sigma} = 0 \\
+ &{u}_{K,\sigma} = {u}_{L,\sigma} = {u}_{\sigma}.
+ \label{eq:sigmaConditions}
+ \end{aligned}
+\end{equation}
+Using these conditions, the intermediate face unknowns ${u}_\sigma$ can be eliminated and the fluxes are expressed as a function of the cell unknowns $u_N$ and associated transmissibilities $t^N_{K,\sigma}$:
+
+\begin{equation}
+ F_{K,\sigma} = \sum_{N \in \mathcal{S}_{K,\sigma}} t^N_{K,\sigma} u_{N}.
+ \label{eq:FVFluxExpression}
+\end{equation}
+
+\begin{figure} [ht]
+\centering
+\includegraphics[width=0.8\linewidth,keepaspectratio]{png/mpfa_iv.png}
+\caption{Interaction region for the Mpfa-O method. The graphic on the right illustrates how the sub-control volume $L^v$ and face $\sigma^v_2$ are embedded in cell $L$. Note that the face stencils for all sub-control volume faces in the depicted interaction region are $\mathcal{S}_{\sigma^v_i} = \{ K,L,M \}$, meaning that the fluxes over the sub-control volume faces depend on the three cell unknowns $u_K, u_L, u_M$.}
+\label{pc:interactionRegion_mpfa}
+\end{figure}
+
+The main difference between the various finite-volume schemes available is the assembly of the face fluxes, i.e. the computation of the $t^N_{K,\sigma}$ and the size of $\mathcal{S}_{K,\sigma}$. For the Tpfa, that has been presented in the last section, the stencil and transmissibilities are given as
+\begin{equation*}
+\mathcal{S}_{K,\sigma} = \lbrace K,L \rbrace, \quad t^K_{K,\sigma} = \meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}},\; t^L_{K,\sigma} = -\meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}},
+\end{equation*}
+with $t_{K,\sigma},t_{L,\sigma}$ as defined in equation \eqref{eq:conormalDecTpfa}.
+
+In the following, a multi-point flux approximation method (Mpfa-O method), which was introduced in \citet{A3:aavatsmark:2002}, is presented. The main difference to the Tpfa scheme is the fact that a consistent discrete gradient is constructed, i.e. the term $\nabla u \cdot \mathbf{d}^{\bot}_{K,\sigma}$ is not neglected.
+
+For this scheme, a dual grid is created by connecting the barycenters of the cells with the barycenters of the faces ($d=2$) or the barycenters of the faces and edges ($d=3$). This divides each cell into sub-control volumes $K^v$. Analogously, each face is sub-divided into sub-control volume faces $\sigma^v$, see Figure \ref{pc:interactionRegion_mpfa}. We allow for piecewise constant $\mathbf{\Lambda}$ (denoted as $\mathbf{\Lambda}_K$ for each cell $K$) and construct discrete gradients $\nabla_\mathcal{D}^{K^v} u$ (per sub-control volume $K^v$).
+In the following, we restrict our discussion to the two-dimensional setup that is shown in Figure \ref{pc:interactionRegion_mpfa}.
+Here, the discrete gradients are constructed to be consistent such that the following conditions hold:
+\begin{equation}
+\nabla_\mathcal{D}^{K^v} u \cdot (\mathbf{x}_{\sigma^v_1}- \mathbf{x}_{K}) = u_{\sigma^v_1} - u_K, \quad \nabla_\mathcal{D}^{K^v} u \cdot (\mathbf{x}_{\sigma^v_3}- \mathbf{x}_{K}) = u_{\sigma^v_3} - u_K.
+\end{equation}
+Thus, a discrete gradient (for sub-control volume $K^v$) that fulfills these conditions is given as
+\begin{equation}
+\nabla_\mathcal{D}^{K^v} u = \mathbb{D}^{-T}_{K^v}
+ \begin{bmatrix}
+ u_{\sigma^v_1} - u_K \\
+ u_{\sigma^v_3} - u_K
+ \end{bmatrix}, \qquad \text{ with }\; \mathbb{D}_{K^v} :=
+ \begin{bmatrix}
+ \mathbf{x}_{\sigma^v_1}- \mathbf{x}_K & \mathbf{x}_{\sigma^v_3} - \mathbf{x}_K
+ \end{bmatrix}.
+ \label{eq:MPFAGradientRecons}
+\end{equation}
+
+This enables us to write the discrete flux across $\sigma^v_1$ from cell $K$ as follows:
+\begin{equation}
+ F_{K, \sigma^v_1} := - |\sigma^v_1| \mathbf{n}_{\sigma^v_1}^T \mathbf{\Lambda}_K \nabla_\mathcal{D}^{K^v} u.
+ \label{eq:discreteFlux}
+\end{equation}
+Inserting the discrete gradient, yields
+\begin{equation}
+ F_{K, \sigma^v_1} = \omega_{K,\sigma^v_1\sigma^v_1}(u_K - u_{\sigma^v_1}) + \omega_{K,\sigma^v_1 \sigma^v_3}(u_K - u_{\sigma^v_3}),
+ \label{eq:discreteFluxRef}
+\end{equation}
+with $(\omega_{K,\sigma^v_1\sigma^v_1},\omega_{K,\sigma^v_1 \sigma^v_3})^T = |\sigma^v_1| \mathbb{D}^{-1}_{K^v}\mathbf{\Lambda}_K \mathbf{n}_{\sigma^v_1}$. These values are calculated in \Dumux by using the function \texttt{computeMpfaTransmissibility}.
+\\ \ \\
+To deduce a cell-centered scheme, the introduced face unknowns $u_{\sigma^v_i}$ have to be eliminated. This is done by enforcing flux continuity for each sub-control volume face, i.e.
+\begin{align}
+F_{K, \sigma^v_1} + F_{L, \sigma^v_1} &= 0, \\ F_{K, \sigma^v_3} + F_{M, \sigma^v_3} &= 0, \\ F_{L, \sigma^v_2} + F_{M, \sigma^v_2} &= 0.
+\end{align}
+This results in a system of equations for the face unknowns $\mathbf{u}_{\sigma}$
+\begin{equation}
+\mathbb{A}^{3\times 3} \mathbf{u}_{\sigma} = \mathbb{B}^{3\times 3} \mathbf{u},
+\end{equation}
+where $\mathbf{u}$ contains the three cell unknowns $u_K,u_L,u_M$ and $\mathbf{u}_{\sigma}$ the three face unknowns $u_{\sigma^v_1}, u_{\sigma^v_2}, u_{\sigma^v_3}$.
+Inserting these face unknowns into the flux expression \eqref{eq:discreteFluxRef} yields
+\begin{equation}
+ F_{K,\sigma^v_i} = \sum_{N \in \lbrace K,L,M \rbrace } t^N_{K,\sigma^v_i} u_{N} = \mathbf{t}_{K,\sigma^v_i} \cdot \mathbf{u},
+ \label{eq:FVFluxExpressionSubFace}
+\end{equation}
+for each cell $K$ and sub-control volume face $\sigma^v_i$.
+
% \subsubsection{NLTPFA}\label{cc_nltpfa}
% TODO
diff --git a/doc/handbook/CMakeLists.txt b/doc/handbook/CMakeLists.txt
index 01cf2ee849d268170050adcb9ce85ef115e4b8d2..5b3a54ea25982e5640bbc939e4e6cf579352b358 100644
--- a/doc/handbook/CMakeLists.txt
+++ b/doc/handbook/CMakeLists.txt
@@ -23,13 +23,14 @@ set(TEX_INPUTS
set(TEX_IMAGES
png/box_disc.png
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+ png/cctpfa.png
png/dumux_strucutre_flowchart_horizontal_explained.png
png/dunedesign.png
../logo/dumux_logo_hires_whitebg.png
png/dalton1.png
png/dalton2.png
- pdf/staggered_grid.pdf)
+ pdf/staggered_grid.pdf
+ png/mpfa_iv.png)
dune_add_latex_document(0_dumux-handbook.tex
BIBFILES dumux-handbook.bib
diff --git a/doc/handbook/png/cctpfa.png b/doc/handbook/png/cctpfa.png
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diff --git a/doc/handbook/png/mpfa_iv.png b/doc/handbook/png/mpfa_iv.png
new file mode 100644
index 0000000000000000000000000000000000000000..a54765dd8792a7e66b783696048467143c90cdec
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diff --git a/doc/handbook/svg/cctpfa.svg b/doc/handbook/svg/cctpfa.svg
new file mode 100644
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+ style="font-size:1.25px;line-height:1.25"
+ y="521.48505"
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+ sodipodi:role="line">$u_3$</tspan></text>
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+ y="546.00702"
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+ style="font-size:1.25px;line-height:1.25"
+ y="546.00702"
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+ sodipodi:role="line">$\bar{u}_{\sigma_1}$</tspan></text>
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+ y="536.96051"
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+ style="font-style:normal;font-weight:normal;line-height:0%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ xml:space="preserve"
+ sodipodi:linespacing="0%"><tspan
+ style="font-size:1.25px;line-height:1.25"
+ y="536.96051"
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+ sodipodi:role="line">$\bar{u}_{\sigma_2}$</tspan></text>
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+ xml:space="preserve"
+ sodipodi:linespacing="0%"><tspan
+ style="font-size:1.25px;line-height:1.25"
+ y="553.08307"
+ x="286.09006"
+ id="tspan12563-7-7-7-6-3"
+ sodipodi:role="line">\tiny $q$</tspan></text>
+ <text
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+ y="571.84045"
+ x="356.35568"
+ style="font-style:normal;font-weight:normal;line-height:0%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ xml:space="preserve"
+ sodipodi:linespacing="0%"><tspan
+ style="font-size:1.25px;line-height:1.25"
+ y="571.84045"
+ x="356.35568"
+ id="tspan12563-7-2-9-2"
+ sodipodi:role="line">sub-control volume</tspan></text>
+ <text
+ id="text12561-8-6-0-0-6"
+ y="584.7547"
+ x="336.60547"
+ style="font-style:normal;font-weight:normal;line-height:0%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ xml:space="preserve"
+ sodipodi:linespacing="0%"><tspan
+ style="font-size:1.25px;line-height:1.25"
+ y="584.7547"
+ x="336.60547"
+ id="tspan12563-7-2-9-6-1"
+ sodipodi:role="line">sub-control volume face</tspan></text>
+ <text
+ id="text12561-8-6-9-8"
+ y="598.94989"
+ x="324.27731"
+ style="font-style:normal;font-weight:normal;line-height:0%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ xml:space="preserve"
+ sodipodi:linespacing="0%"><tspan
+ style="font-size:1.25px;line-height:1.25"
+ y="598.94989"
+ x="324.27731"
+ id="tspan12563-7-2-3-7"
+ sodipodi:role="line">interaction region</tspan></text>
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+ id="text12561-8-6-9-6-9"
+ y="491.0112"
+ x="308.02261"
+ style="font-style:normal;font-weight:normal;line-height:0%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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