Fix formula mark-up.

diff --git a/advection_reaction_estimator/README.md b/advection_reaction_estimator/README.md
index 65558e534bd768b7aa2008aa43e5038a95c2ec53..37879403e215603e406e802e44937272d6207be5 100644 (file)
--- a/advection_reaction_estimator/README.md
+++ b/advection_reaction_estimator/README.md
@@ -23,10 +23,19 @@ If you run `./DG_advection_reaction parameters.prm`, an error message will tell
 ## The problem:
 This program solves the problem, for $\Omega \in \mathbb{R^2}$
 
-$$\begin{cases} b \cdot \nabla u + c u = f \qquad  \text{in } \Omega \\
-\qquad \qquad u=g \qquad \text{on } \partial_{-}\Omega \end{cases}$$
+@f[
+\begin{cases} b \cdot \nabla u + c u = f \qquad  \text{in } \Omega \\
+\qquad \qquad u=g \qquad \text{on } \partial_{-}\Omega \end{cases}
+@f]
 
-where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in \partial \Omega: b(x)\cdot n(x) <0\}$ is the inflow part of the boundary, with $b=(b_1,b_2) \in \mathbb{R^2}$. As we know from classical DG theory, we need to ensure that $$c(x) - \frac{1}{2}\nabla \cdot b \geq \gamma_0 >0$$for some positive $\gamma_0$ so that we have coercivity in $L^2$ at the continuous level. Discrete coercivity is achieved by using a stronger norm which takes care of jumps, see Di-Pietro Ern [1] for details.
+where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in
+\partial \Omega: b(x)\cdot n(x) <0\}$ is the inflow part of the
+boundary, with $b=(b_1,b_2) \in \mathbb{R^2}$. As we know from
+classical DG theory, we need to ensure that 
+@f[
+c(x) - \frac{1}{2}\nabla \cdot b \geq \gamma_0 >0
+@f]
+for some positive $\gamma_0$ so that we have coercivity in $L^2$ at the continuous level. Discrete coercivity is achieved by using a stronger norm which takes care of jumps, see Di-Pietro Ern [1] for details.
 
 
 ## The weak formulation:
@@ -35,37 +44,52 @@ where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in \partial
 
 As trial space we choose $V_h = \{ v_h \in L^2(\Omega): v_h \in P^1(\mathbb{T_h})\} \notin H^1(\Omega)$. If we integrate by parts and sum over all cells
 
-$$\sum_{T \in \mathbb{T}_h} \Bigl( (-u,\beta \cdot \nabla v_h) _T + (c u,v_h)_T + \bigl<(b \cdot n) u ,v_h \bigr>_{\partial T} \Bigr) = (f,v_h)_{\Omega}$$
+@f[
+\sum_{T \in \mathbb{T}_h} \Bigl( (-u,\beta \cdot \nabla v_h) _T + (c
+u,v_h)_T + \bigl<(b \cdot n) u ,v_h \bigr>_{\partial T} \Bigr) =
+(f,v_h)_{\Omega}
+@f]
 
 and use the so-called DG magic formula and exploit the property $[bu]_{\mathbb{F}^i} = 0$ where $\mathbb{F}^i$ are set of internal faces we obtain the (unstable!) formulation:
 
 Find $u_h \in V_h$: 
 
-$$
+@f[
     a_h(u_h,v_h) + b_h(u_h,v_h)=l(v_h) \qquad \forall v_h \in V_h
-$$
+@f]
 where
-$$
+@f[
 a_h(u,v_h)=\sum_{T \in \mathbb{T}_h} \Bigl( (-u,b \cdot \nabla v_h) _T + (c u,v_h)_T \Bigr)
-$$
+@f]
 
-$$    b_h(u,v_h)= \sum_{F \not \in \partial_{-}\Omega} \bigl< \{ b u\}, [v_h]\bigr>_F $$
+@f[    
+b_h(u,v_h)= \sum_{F \not \in \partial_{-}\Omega} \bigl< \{ bu\}, [v_h]\bigr>_F 
+@f]
 
-$$
+@f[
     l(v_h)= (f,v_h)_{\Omega} - \sum_{F \in \partial_{-}\Omega} \bigl< (b \cdot n) g,v_h \bigr>_F
-$$
+@f]
 
 It's well known this formulation is coercive only in $L^2$, hence the formulation is unstable as we don't "see" the derivatives. To stabilize this, we can use a jump-penalty term, i.e. our $b_h$ is replaced by:
 
-$$b_h^s(u_h,v_h)=b_h(u_h,v_h)+ \sum_{F \in \mathbb{F}^i} \bigl< c_F [u_h],[v_h]  \bigr> $$
+@f[
+b_h^s(u_h,v_h)=b_h(u_h,v_h)+ \sum_{F \in \mathbb{F}^i} \bigl< c_F
+[u_h],[v_h]  \bigr> 
+@f]
 
 where $c_F>0$ is a function on each edge such that $c_F \geq \theta |b \cdot n|$ for some positive $\theta$. In this program, $\theta=\frac{1}{2}$ and $c_F = \frac{1}{2} |b \cdot n|$, which corresponds to an upwind formulation. Notice that consistency is trivially achieved, as $[u]_{\mathbb{F}^i} =0$. This formulation is stable in the energy norm 
 
-$$    |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2 \Bigr)^{\frac{1}{2}}$$
+@f[
+    |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in
+    \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2
+    \Bigr)^{\frac{1}{2}}
+@f]
 
 (well defined on $H^1(\Omega) + V_h$) and moreover we have the a-priori bound:
 
-$$|||u-u_h||| \leq C h^{k+\frac{1}{2}}||u||_{k+1,\Omega} $$
+@f[
+|||u-u_h||| \leq C h^{k+\frac{1}{2}}||u||_{k+1,\Omega} 
+@f]
 
 valid for $u \in H^{k+1}(\Omega)$.
 
@@ -79,7 +103,11 @@ The estimator is the one proposed by Georgoulis, Edward Hall and Charalambos Mak
 
 The reliability is:
 
-$$|||u-u_h|||^2 \leq  C || \sqrt{b \cdot n}[u_h]||_{\Gamma^{-}}^2 + C \sum_{T \in \mathbb{T}_h}\Bigl( ||\beta (g-u_h^+)||_{\partial_{-}T \cap \partial_{-} \Omega}^2 +||f-c u_h - \Pi(f- cu_h)||_T^2 \Bigr)$$
+@f[
+|||u-u_h|||^2 \leq  C || \sqrt{b \cdot n}[u_h]||_{\Gamma^{-}}^2 + C
+\sum_{T \in \mathbb{T}_h}\Bigl( ||\beta (g-u_h^+)||_{\partial_{-}T
+\cap \partial_{-} \Omega}^2 +||f-c u_h - \Pi(f- cu_h)||_T^2 \Bigr)
+@f]
 
 where:
 
@@ -119,4 +147,4 @@ If we look at the decrease of the energy norm of the error in the globally refin
 * [1] Emmanuil H. Georgoulis, Edward Hall and Charalambos Makridakis (2013), Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems. DOI: [10.1007/978-3-319-01818-8_8
 ](https://link.springer.com/chapter/10.1007%2F978-3-319-01818-8_8)
 * [2] Di Pietro, Daniele Antonio and Ern, Alexandre (2012), Mathematical Aspects of Discontinuous Galerkin Methods. ISBN: [978-3-642-22980-0](https://www.springer.com/gp/book/9783642229794)
-* [3] Franco Brezzi, Luisa Donatella Marini and Endre Süli (2004) Discontinuous Galerkin Methods for First-Order Hyperbolic Problems. DOI: [10.1142/S0218202504003866](https://doi.org/10.1142/S0218202504003866)
\ No newline at end of file
+* [3] Franco Brezzi, Luisa Donatella Marini and Endre Süli (2004) Discontinuous Galerkin Methods for First-Order Hyperbolic Problems. DOI: [10.1142/S0218202504003866](https://doi.org/10.1142/S0218202504003866)