Correct mathjax rendering for injection.html

Author: tianshi-liu

projects/injection.html

@@ -1,38 +1,47 @@
-<script src="https://rawcdn.githack.com/oscarmorrison/md-page/232e97938de9f4d79f4110f6cfd637e186b63317/md-page.js"></script><noscript>
+<script type="text/x-mathjax-config">
+    MathJax.Hub.Config({
+      tex2jax: {
+        inlineMath: [ ['$','$'], ["\\(","\\)"] ],
+        processEscapes: true
+      }
+    });
+</script>
+
 <script
   src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"
   type="text/javascript">
 </script>
 
-# Wavefield injection based on interface discontinuity conditions
-## Definition of the wavefield injection problem
-We define and discuss the wavefield injection problem in the context of linear elasticity. The wavefield is governed by the elastic wave equation
-
+<h1> Wavefield injection based on interface discontinuity conditions</h1>
+<h2> Definition of the wavefield injection problem</h2>
+<p>We define and discuss the wavefield injection problem in the context of linear elasticity. The wavefield is governed by the elastic wave equation
 $$\rho\frac{\partial^2 \mathbf{u}}{\partial t^2}-\nabla\cdot(\mathbf{C}:\nabla\mathbf{u}) = \mathbf{f}\qquad \text{in}\ \oplus,$$
+subject to certain boundary condition at $\partial\oplus$, where $\oplus$ denotes the entire medium of wave propagation (e.g., the whole Earth). </p>
 
-subject to certain boundary condition at $\partial\oplus$, where $\oplus$ denotes the entire medium of wave propagation (e.g., the whole Earth). 
-
-Now suppose that we have a *local domain* $\Omega\in\oplus$ in which structural perturbations exist and the wave propagations are of particular interest, but the source $\mathbf{f}$ is supported far away from $\Omega$, and directly solving the wavefield in $\oplus$ is too expensive. If we happen to have access to some *background wavefield* $\mathbf{u}^0$ outside $\Omega$
-
+<p>Now suppose that we have a <em>local domain</em> $\Omega\in\oplus$ in which structural perturbations exist and the wave propagations are of particular interest, but the source $\mathbf{f}$ is supported far away from $\Omega$, and directly solving the wavefield in $\oplus$ is too expensive. If we happen to have access to some <em>background wavefield</em> $\mathbf{u}^0$ outside $\Omega$
+  
 $$\rho^0\frac{\partial^2 \mathbf{u}^0}{\partial t^2}-\nabla\cdot(\mathbf{C}^0:\nabla\mathbf{u}) = \mathbf{f}\qquad \text{in}\ \oplus\setminus\Omega,$$
+  
+then it may be possible to reconstruct $\mathbf{u}$ using $\mathbf{u}^0$, with numerical simulations only performed in $\Omega$ (potentially in the neighbourhood of $\Omega$, but not in the entire $\oplus$).</p>
 
-then it may be possible to reconstruct $\mathbf{u}$ using $\mathbf{u}^0$, with numerical simulations only performed in $\Omega$ (potentially in the neighbourhood of $\Omega$, but not in the entire $\oplus$).
 
-As a more rigorous definition, if
-1. there is no structural perturbations in the local domain
+<p>As a more rigorous definition, if
+<ol>
+<li> there is no structural perturbations in the local domain
 
 $$\rho = \rho^0, \qquad \mathbf{C} = \mathbf{C}^0 \qquad \text{in}\ \overline{\oplus\setminus\Omega},$$
 
-2. the source term is supported outside the local domain
+<li> the source term is supported outside the local domain
 
 $$\mathbf{f} = 0 \qquad \text{in}\ \overline{\Omega},$$
+</ol>
 
-then reconstructing $\mathbf{u}$ using $\mathbf{u}^0$ is called the *wavefield injection problem*.
+then reconstructing $\mathbf{u}$ using $\mathbf{u}^0$ is called the <em>wavefield injection problem</em>.</p>
 
-## The equivalence between the wavefield injection problem and the interface discontinuity condition problem
-If we define wavefield
+<h2> The equivalence between the wavefield injection problem and the interface discontinuity condition problem</h2>
+<p>If we define wavefield
 
-$$\mathbf{w}:=\left\\{\begin{array}{ll}
+$$\mathbf{w}:=\left\{\begin{array}{ll}
 \mathbf{u} & \qquad \text{in}\ \Omega, \\
 \mathbf{u}-\mathbf{u}^0   & \qquad \text{in}\ \oplus\setminus\Omega,
 \end{array}\right.$$
@@ -41,8 +50,8 @@
 
 $$\rho\frac{\partial^2 \mathbf{w}}{\partial t^2}-\nabla\cdot(\mathbf{C}:\nabla\mathbf{w}) = 0\qquad \text{in}\ \oplus,$$
 
-but $\mathbf{w}$ is discontinuous in its displacement and traction across the interface between $\Omega$ and $\oplus\setminus\Omega$
+but $\mathbf{w}$ is discontinuous in its displacement and traction across the interface between $\Omega$ and $\oplus\setminus\Omega$:
 
-$$\mathbf{w}\Big|\_{\Gamma+}^{\Gamma-} = \mathbf{u}^0\Big|\_{\Gamma},$$
+$$\mathbf{w}\Big|_{\Gamma+}^{\Gamma-} = \mathbf{u}^0\Big|_{\Gamma},$$
 
-$$\hat{\mathbf{n}}\cdot(\mathbf{C}:\nabla\mathbf{w})\Big|\_{\Gamma+}^{\Gamma-} = \hat{\mathbf{n}}\cdot(\mathbf{C}:\nabla\mathbf{u}^0)\Big|\_{\Gamma}.$$
+$$\hat{\mathbf{n}}\cdot(\mathbf{C}:\nabla\mathbf{w})\Big|_{\Gamma+}^{\Gamma-} = \hat{\mathbf{n}}\cdot(\mathbf{C}:\nabla\mathbf{u}^0)\Big|_{\Gamma}.$$</p>