diff --git a/slides/intro.md b/slides/intro.md
index 2b3b294ce4affcbe81698eb232dd941417eab8e6..fcd73c679c673bad99df8f664d9ff4e591053d51 100644
--- a/slides/intro.md
+++ b/slides/intro.md
@@ -37,7 +37,7 @@ subtitle: Overview and Available Models
 
 ## Overview
 
-<img src="img/dumux.png" width="400"/>
+<img src="img/dumux.png" width="300"/>
 
 * **DuMu^x^:** DUNE for Multi-{Phase, Component, Scale, Physics, $\text{...}$} flow and transport in porous media.
 * **Goal:** **sustainable, consistent, research-friendly framework** for the implementation and application of
@@ -81,7 +81,7 @@ and third-party funding aquired at the LH^2^
 
 ## Funding
 
-We acknowledge funding that supported the development of DuMu^x^ from
+We acknowledge funding that supported the development of DuMu^x^ in past and present:
 
 <img src="img/funding.svg" width="550"/>
 
@@ -115,17 +115,19 @@ We acknowledge funding that supported the development of DuMu^x^ from
 * DuMu^x^ **Examples** (<https://git.iws.uni-stuttgart.de/dumux-repositories/dumux/-/tree/master/examples#examples>)
 * DuMu^x^ **Website** (<https://dumux.org/>)
 
-# Available Models
+# Mathematical Models
 
 ## Mathematical Models
 
-* **Porous medium flow (Darcy)**: Single and multi-phase models for flow and transport in porous materials.
-* **Free flow (Navier-Stokes)**: Single-phase models based on the Navier-Stokes equation.
+Preimplemented models:
+
+* **Flow in porous media (Darcy)**: Single and multi-phase models for flow and transport in porous materials.
+* **Free flow (Navier-Stokes)**: Single-phase models based on the Navier-Stokes equations.
 * **Shallow water flow**: Two-dimensional shallow water flow (depth-averaged).
-* **Geomechanics**: Models taking into account solid deformation.
+* **Geomechanics**: Models taking into account solid deformation of porous materials.
 * **Pore network**: Single and multi-phase models for flow and transport in pore networks.
 
-## Available Models
+## Flow in Porous Media
 
 <img src="img/models.png" width="650"/>
 
@@ -140,13 +142,14 @@ We acknowledge funding that supported the development of DuMu^x^ from
 * Multi-phase flow (phase $\alpha$)
 
     $v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right)$
+    where $k_{r\alpha}(S_\alpha)$ is the relative permeability, a function of saturation $S_\alpha$
 
 ## 1p -- single-phase
 
 * Uses standard Darcy approach for the conservation of momentum
 * Mass continuity equation
 
-    $\phi \frac{\partial \varrho}{\partial t} + \text{div} \left\lbrace - \varrho \frac{\textbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) \right\rbrace = q$
+    $\frac{\partial\left( \phi \varrho \right)}{\partial t} + \text{div} \left\lbrace - \varrho \frac{\textbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) \right\rbrace = q$
 
 * Primary variable: $p$
 
@@ -155,7 +158,7 @@ We acknowledge funding that supported the development of DuMu^x^ from
 * Uses standard Darcy approach for the conservation of momentum
 * Transport of component $\kappa \in \{w, a, ...\}$
 
-    $\phi \frac{\partial \varrho X^\kappa}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa \frac{\textbf {K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q$
+    $\frac{\partial\left( \phi \varrho X^\kappa \right)}{\partial t} - \text{div} \left\lbrace \varrho X^\kappa \frac{\textbf {K}}{\mu} \left(\textbf{grad}\, p - \varrho \textbf{g} \right) + \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right\rbrace = q$
 
 * Primary variables: $p$ and $x^\kappa$
 
@@ -178,69 +181,75 @@ We acknowledge funding that supported the development of DuMu^x^ from
 * Uses standard multi-phase Darcy approach for the conservation of momentum
 * Conservation of the phase mass of phase $\alpha \in \{w, n\}$
 
-    $\phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{\varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} = q_\alpha$
+    $\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} - \text{div} \left\{\varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\} = q_\alpha$
 
-* Constitutive relation $p_c = p_n - p_w$
-* $S_w + S_n = 1$
-* Primary variables: $p_w$ and $S_n$ or $p_n$ and $S_w$
+* Constitutive relation: $p_c := p_n - p_w = p_c(S_w)$, $k_{r\alpha}$ = $k_{r\alpha}(S_w)$
+* Physical constraint (no free space): $S_w + S_n = 1$
+* Primary variables: $p_w$, $S_n$ or $p_n$, $S_w$
 
 ## 2pnc
 
 * Transport equation for each component $\kappa \in \{w, n, ...\}$ in phase $\alpha \in \{w, n\}$
 
-    $\frac{\partial \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha}{\partial t} - \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace$
-    $- \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa$
+    $\begin{aligned}\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &- \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace \\
+    &- \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa \end{aligned}$
 
-* Constitutive relation $p_c = p_n - p_w$
-* $S_w + S_n = 1$ and $X^\kappa_w + X^\kappa_n = 1$
-* Primary variables: depend on the phase state
+* Constitutive relation: $p_c := p_n - p_w = p_c(S_w)$, $k_{r\alpha}$ = $k_{r\alpha}(S_w)$
+* Physical constraints: $S_w + S_n = 1$ and $\sum_\kappa X_\alpha^\kappa = 1$
+* Primary variables: depending on the phase state
 
 ## 2pncmin
 
 * Transport equation for each component $\kappa \in \{w, n, ...\}$ in phase $\alpha \in \{w, n\}$
 
-    $\frac{\partial \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha}{\partial t} - \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace$
-    $- \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa$
+    $\begin{aligned}\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &- \sum_\alpha \text{div} \left\lbrace \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left( \textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace \\
+    &- \sum_\alpha \text{div} \left\lbrace \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right\rbrace = \sum_\alpha q_\alpha^\kappa \end{aligned}$
 
 * Mass balance solid or mineral phases
 
-    $\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda$
+    $\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda \quad \forall \lambda \in \Lambda$
 
-* $p_c = p_n - p_w$, $S_w + S_n = 1$ and $X^\kappa_w + X^\kappa_n = 1$
-* Primary variables: depend on the phase state
+* for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
+* source term models **dissolution** / **precipiation** / **phase transition** fluid &harr; solid
 
 ## 3p -- three-phase
 
 * Uses standard multi-phase Darcy approach for the conservation of momentum
 * Conservation of the phase mass of phase $\alpha \in \{w, g, n\}$
 
-    $\phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace = q_\alpha$
+    $\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} - \text{div} \left\lbrace \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace = q_\alpha$
 
-* $S_w + S_n + S_g = 1$
-* Primary variables: $p_g$, $S_w$ and $S_n$
+* Physical constraint: $S_w + S_n + S_g = 1$
+* Primary variables: $p_g$, $S_w$, $S_n$
 
 ## 3p3c
 
 * Transport equation for each component $\kappa \in \{w, a, c\}$ in phase $\alpha \in \{w, g, n\}$
 
-    $\phi \frac{\partial \left(\sum_\alpha \varrho_{\alpha,mol} x_\alpha^\kappa S_\alpha \right)}{\partial t}$
-    $- \sum_\alpha \text{div} \left\lbrace \frac{k_{r\alpha}}{\mu_\alpha} \varrho_{\alpha,mol} x_\alpha^\kappa \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha,mass} \mathbf{g} \right) \right\rbrace$
-    $- \sum_\alpha \text{div} \left\lbrace D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right\rbrace = q^\kappa$
+    $\begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_{\alpha,mol} x_\alpha^\kappa S_\alpha \right)}{\partial t}
+    &- \sum_\alpha \text{div} \left\lbrace \frac{k_{r\alpha}}{\mu_\alpha} \varrho_{\alpha,mol} x_\alpha^\kappa \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha,mass} \mathbf{g} \right) \right\rbrace \\
+    &- \sum_\alpha \text{div} \left\lbrace D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right\rbrace = q^\kappa \end{aligned}$
 
-* $S_w + S_n + S_g = 1$ and $x^w_\alpha + x^a_\alpha + x^c_\alpha = 1$
+* Physical constraints: $\sum_\alpha S_\alpha = 1$ and $\sum_\kappa x^\kappa_\alpha = 1$
 * Primary variables: depend on the locally present fluid phases
 
-## Non-Isothermal
-
-* Local thermal equilibrium is assumed
+## Non-Isothermal (equilibrium)
 
+* Local thermal equilibrium assumption
 * One energy conservation equation for the porous solid matrix and the fluids
 
-    $\phi \frac{\partial \sum_\alpha \varrho_\alpha u_\alpha S_\alpha}{\partial t} + \left(1 - \phi \right) \frac{\partial \left(\varrho_s c_s T \right)}{\partial t}$
-    $- \sum_\alpha \text{div} \left\lbrace \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace$
-    $- \text{div} \left(\lambda_{pm} \textbf{grad}\, T \right) = q^h$
+    $\begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} &+ \frac{\partial \left(\left(1 - \phi \right)\varrho_s c_s T \right)}{\partial t} \\
+    &- \sum_\alpha \text{div} \left\lbrace \varrho_\alpha h_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right) \right\rbrace \\
+    &- \text{div} \left(\lambda_{pm} \textbf{grad}\, T \right) = q^h \end{aligned}$
+
+* specific internal energy $u_\alpha = h_\alpha - p_\alpha / \varrho_\alpha$
+* can be added to other models, additional primary variable temperature $T$
 
-* $u_\alpha = h_\alpha - p_\alpha / \varrho_\alpha$
+## Free flow (Navier-Stokes)
+
+* Stokes equation
+* Navier-Stokes equations
+* Energy and component transport
 
 ## Reynolds-Averaged Navier-Stokes (RANS)
 
@@ -253,14 +262,16 @@ We acknowledge funding that supported the development of DuMu^x^ from
 
     $\mu_\textrm{eff} = \mu + \mu_\textrm{t}$
 
+* Various turbulence models are implemented
+
 # Spatial Discretization
 
-## Cell Centered Finite Volume Methods
+## Cell-centered Finite Volume Methods
 
-* Use elements of the grid as control volumes
-* Discrete **values** are determined at the element/control volume **center**
+* Elements of the grid are used as control volumes
+* Discrete **values** represent control volume average
 * **Two-point flux approximation (TPFA)**
-    * Simple but robust
+    * Simple and robust but not always consistent
 * **Multi-point flux approximation (MPFA)**
     * A consistent discrete gradient is constructed
 
@@ -272,28 +283,31 @@ We acknowledge funding that supported the development of DuMu^x^ from
 
 <img src="img/mpfa.png" width="80%"/>
 
-## Box method
+## Control-volume finite element methods
 
 * Model domain is discretized using a **FE** mesh
 * Secondary **FV** mesh is constructed &rarr; control volume/**box**
-* Control volumes are partitioned into sub-control volumes (scvs)
-* Faces of control volumes are partitioned into sub-control volume faces (scvfs)
-* Unites advantages of finite-volume and finite-element methods
+* Control volumes (CV) split into sub control volumes (SCVs)
+* Faces of CV split into sub control volume faces (SCVFs)
+* Unites advantages of finite-volume (simplicity) and finite-element methods (flexibility)
     * **Unstructured grids** (from FE method)
-    * **Mass conservative** (from FV method)
+    * **Mass conservation** (from FV method)
 
 ## Box method
 
-<img src="img/box.png"/>
+Vertex-centered finite volumes / control volume finite element
+method with piecewise linear polynomial functions ($\mathrm{P}_1/\mathrm{Q}_1$)
+
+<img src="img/box.png" width="70%"/>
 
-## Staggered Grid
+## Finite Volume method on staggered grid
 
-* Uses a finite volume method with different control volumes for different equations
+* Uses a finite volume method with different staggered control volumes for different equations
 * Fluxes are evaluated with a two-point flux approximation
 * **Robust** and **mass conservative**
-* Should be applied for **structured grids** only
+* Restricted to **structured grids** (tensor-product structure)
 
-## Staggered Grid
+## Staggered grid discretization
 
 <img src="img/staggered_grid.png"/>
 
@@ -301,15 +315,14 @@ We acknowledge funding that supported the development of DuMu^x^ from
 
 ## Model Components
 
-* The following components have to be specified
-    * **Solver**: Type of solution stategy
-    * **Assembler**: Key properties
-        * Geometry, Variables, LocalResidual
-    * **LinearSolver**: How to solve algebraic equations
-    * **Problem**: Initial and boundary conditions
-    * **SolutionVector**: Container to store the solution
+* Typically, the following components have to be specified
+    * **Model**: Equations and constitutive models
+    * **Assembler**: Key properties (Discretization, Variables, LocalResidual)
+    * **Solver**: Type of solution stategy (e.g. Newton)
+    * **LinearSolver**: Method for solving linear equation systems (e.g. direct / Krylov subspace methods)
+    * **Problem**: Initial and boundary conditions, source terms
     * **TimeLoop**: For time-dependent problems
-    * **IOFields** and **VtkOutputModule**: Output of the simulation
+    * **VtkOutputModule** / **IOFields**: For VTK output of the simulation
 
 # Simulation Flow