From 812e06e294de908186b50d823e3c13435f0a6e95 Mon Sep 17 00:00:00 2001
From: Timo Heister <timo.heister@gmail.com>
Date: Tue, 14 May 2019 11:43:28 -0600
Subject: [PATCH] small fixes

---
 examples/step-63/doc/intro.dox | 10 +++++-----
 examples/step-63/step-63.cc    | 23 +++++++++++------------
 2 files changed, 16 insertions(+), 17 deletions(-)

diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox
index 61f5df839c..f12dcad9a9 100644
--- a/examples/step-63/doc/intro.dox
+++ b/examples/step-63/doc/intro.dox
@@ -35,7 +35,7 @@ direction</i>, and $f$ is a source. A few notes:
 2. If $\varepsilon=0$ then this is the stationary advection equation solved in
 step-9.
 
-3. Define the \textit{Peclet number}: $\mathcal{P}\coloneqq
+3. Define the <i>Peclet number</i>: $\mathcal{P}\coloneqq
 \|\boldsymbol{\beta}\| \cdot L/\varepsilon$. If $\mathcal{P}>1$, we say the
 problem is <i>advection-dominated</i>, else if $\mathcal{P}<1$ we will
 say the problem is <i>diffusion-dominated</i>. Here $L$ is the length
@@ -68,11 +68,11 @@ numerical solutions when $\varepsilon \ll \|\boldsymbol{\beta}\|\cdot
 L$. To combat this, we will consider the new weak form
 @f{align*}{
 a(u_h,\,v_h) + \sum_K (-\varepsilon \Delta u_h +
-\boldsymbol{\beta}\cdot \nabla u_h-f,\,\delta
-\boldsymbol{\beta}_K\cdot \nabla v_h)_K = F(v_h)
+\boldsymbol{\beta}\cdot \nabla u_h-f,\,\delta_K
+\boldsymbol{\beta}\cdot \nabla v_h)_K = F(v_h)
 @f}
-where the sum is done over all cells $K$ with the inner product is
-understood on that cell, and $\delta_K$ is a cell-wise constant
+where the sum is done over all cells $K$ with the inner product taken
+for each cell, and $\delta_K$ is a cell-wise constant
 stabilization parameter defined in <a
 href="https://link.springer.com/chapter/10.1007/978-3-540-34288-5_27">
 On Discontinuity-Capturing Methods for Convection-Diffusion Equations
diff --git a/examples/step-63/step-63.cc b/examples/step-63/step-63.cc
index 0e061b3f2c..4efa00baad 100644
--- a/examples/step-63/step-63.cc
+++ b/examples/step-63/step-63.cc
@@ -339,17 +339,15 @@ namespace Step63
 
   // @sect3{Right-hand Side and Boundary Values}
 
-  // The problem solved in this tutorial is an adaptation of Ex. 3.1.3
-  // found on pg. 118 of
-  // <a
+  // The problem solved in this tutorial is an adaptation of Ex. 3.1.3 found
+  // on pg. 118 of <a
   // href="https://global.oup.com/academic/product/finite-elements-and-fast-iterative-solvers-9780199678808">
   // Finite Elements and Fast Iterative Solvers: with Applications in
-  // Incompressible Fluid Dynamics by Elman, Silvester, and Wathen</a> with
-  // Applications in Incompressible Fluid Dynamics by Elman, Silvester, and
-  // Wathen. The main difference being that we add a hole in the center of our
-  // domain with zero Dirichlet boundary.
+  // Incompressible Fluid Dynamics by Elman, Silvester, and Wathen</a>. The
+  // main difference being that we add a hole in the center of our domain with
+  // zero Dirichlet boundary.
 
-  // We have a zero right-hand side.
+  // We have a zero right-hand side:
   template <int dim>
   class RightHandSide : public Function<dim>
   {
@@ -448,10 +446,11 @@ namespace Step63
 
   // @sect3{Streamline Diffusion}
 
-  // Streamline diffusion stabilization term. Value is defined in
-  // <a href="https://link.springer.com/chapter/10.1007/978-3-540-34288-5_27">
-  // On Discontinuity—Capturing Methods for Convection—Diffusion Equations
-  // by Volker and Petr</a>
+  // Streamline diffusion stabilization constant. Parameter design is taken
+  // from <a
+  // href="https://link.springer.com/chapter/10.1007/978-3-540-34288-5_27"> On
+  // Discontinuity-Capturing Methods for Convection-Diffusion Equations by
+  // Volker John and Petr Knobloch</a>
   template <int dim>
   double compute_stabilization_delta(const double         hk,
                                      const double         eps,
-- 
2.39.5