fix right test function and Peclet def

diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox
index d18a9fc56ab573ff223829689adffecc46bd533d..61f5df839cd095f0ac38ade124316b0d546be0b0 100644 (file)
--- a/examples/step-63/doc/intro.dox
+++ b/examples/step-63/doc/intro.dox
@@ -35,9 +35,11 @@ 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. If $\varepsilon \ll \|\boldsymbol{\beta}\|$, we say the problem is
-<i>advection-dominated</i>, else we say the problem is
-<i>diffusion-dominated</i>.
+3. Define the \textit{Peclet number}: $\mathcal{P}\coloneqq
+\|\boldsymbol{\beta}\| \cdot L/\varepsilon$. 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
+scale of the domain.
 
 For the discussion in this tutorial we will be concerned with
 advection-dominated flow. 
@@ -49,33 +51,36 @@ a(u_h,v_h) = F(v_h)
 @f}
 where
 @f{align*}{
-a(u_h,v_h) &= (\varepsilon \nabla u_h,\, \nabla v_h) + (\boldsymbol{\beta}\cdot\nabla u_h,\,v_h),\\
-F(v_h) &= (f,\,v_h).
+a(u_h,v_h) &= (\varepsilon \nabla v_h,\, \nabla u_h) +
+(v_h,\,\boldsymbol{\beta}\cdot \nabla u_h),\\
+F(v_h) &= (v_h,\,f).
 @f}
 
 <h3>Streamline diffusion</h3>
 
 The following error estimate can be shown for this PDE:
 @f{align*}{
-\|\nabla (u-u_h)\| \leq (1+\mathcal{P}) \inf_{v_h} \|\nabla (u-v_h)\|
+\|\nabla (u-u_h)\| \leq (1+\mathcal{P}) \inf_{v_h} \|\nabla (u-v_h)\|.
 @f}
-where $\mathcal{P} \sim \|\boldsymbol{\beta}\|/\varepsilon$ (referred
-to as the <i>Peclet</i> number). This implies that we may have poor
-numerical solutions when $\varepsilon \ll \|\boldsymbol{\beta}\|$. To
-combat this, we will consider the new weak form
+
+Given the definition of $\mathcal{P}$ above, we may have poor
+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)
+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)
 @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 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 by Volker
-John and Petr Knobloch</a>. Essentially, adding in the discrete strong form
-residual enhances the coercivity of the bilinear form $a(\cdot,\cdot)$ which
-increases the stability of the discrete solution. This method is commonly
-referred to as <i>streamline diffusion</i> or <i>SUPG</i> (streamline
-upwind/Petrov-Galerkin).
+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
+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
+by Volker John and Petr Knobloch</a>. Essentially, adding in the
+discrete strong form residual enhances the coercivity of the bilinear
+form $a(\cdot,\cdot)$ which increases the stability of the discrete
+solution. This method is commonly referred to as <i>streamline
+diffusion</i> or <i>SUPG</i> (streamline upwind/Petrov-Galerkin).
 
 
 <h2>Smoothers</h2>

diff --git a/examples/step-63/doc/results.dox b/examples/step-63/doc/results.dox
index fe3193f73bfa9b360681e07825ec7e25cdbba3d6..76ca966d88265a91f0a15c0a9adcd46ac1d33fd1 100644 (file)
--- a/examples/step-63/doc/results.dox
+++ b/examples/step-63/doc/results.dox
@@ -20,8 +20,8 @@ The point-wise smoothers ("Jacobi" and "SOR") get applied in the order the
 DoFs are numbered on each level. We can influence this using the
 DoFRenumbering namespace. The block smoothers are applied based on the
 ordering we set in @p setup_smoother. We can visualize this numbering. The
-following picture shows the cell numbering of the active cells in upstream,
-random, and downstream numbering (left to right):
+following picture shows the cell numbering of the active cells in downstream,
+random, and upstream numbering (left to right):
 
 <img src="https://www.dealii.org/images/steps/developer/step-63-cell-order.png" alt="">