From b77eba0414b4e9d2a1d881c8b7112ae4a00b6950 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Thu, 2 May 2024 23:58:13 +0530 Subject: [PATCH] Restate the equation we solve at the beginning of step-6. The equation is the same as step-5, but that is never overly explicitly said. It's also useful to just restate the equations for reference. step-5 does not describe the coefficient we use in the introduction. Add that too while there. --- examples/step-5/doc/intro.dox | 14 +++++++++++--- examples/step-6/doc/intro.dox | 17 ++++++++++++++++- 2 files changed, 27 insertions(+), 4 deletions(-) diff --git a/examples/step-5/doc/intro.dox b/examples/step-5/doc/intro.dox index f7f52f99d7..319aae90b0 100644 --- a/examples/step-5/doc/intro.dox +++ b/examples/step-5/doc/intro.dox @@ -70,9 +70,17 @@ The equation to solve here is as follows: If $a(\mathbf x)$ was a constant coefficient, this would simply be the Poisson equation that we have already solved in step-3 and step-4. However, if it is indeed spatially variable, it is a more complex equation (sometimes referred -to as the "Poisson equation with a coefficient"). Depending on -what the variable $u$ refers to, it models a variety of situations with wide -applicability: +to as the "Poisson equation with a coefficient"). Specifically, we will here +choose it as follows: +@f{align*}{ + a(\mathbf x) = + \begin{cases} + 20 & \text{if}\ |\mathbf x|<0.5, \\ + 1 & \text{otherwise.} + \end{cases} +@f} +Depending on what the variable $u$ refers to, it models a variety of +situations with wide applicability: - If $u$ is the electric potential, then $-a\nabla u$ is the electric current in a medium and the coefficient $a$ is the conductivity of the medium at any diff --git a/examples/step-6/doc/intro.dox b/examples/step-6/doc/intro.dox index c07d309273..408ac00727 100644 --- a/examples/step-6/doc/intro.dox +++ b/examples/step-6/doc/intro.dox @@ -8,7 +8,7 @@ the use of adaptively (locally) refined meshes. The program is still based on step-4 and step-5, and, as you will see, it does not actually take very much code to enable adaptivity. Indeed, while we do a great deal of explaining, adaptive meshes can be added to an existing program -with barely a dozen lines of additional code. The program shows what +with less than ten lines of additional code. The program shows what these lines are, as well as another important ingredient of adaptive mesh refinement (AMR): a criterion that can be used to determine whether it is necessary to refine a cell because the error is large on it, @@ -16,6 +16,21 @@ whether the cell can be coarsened because the error is particularly small on it, or whether we should just leave the cell as it is. We will discuss all of these issues in the following. +The program solves the same problem as step-5, that is, we solve the equation +@f{align*}{ + -\nabla \cdot a(\mathbf x) \nabla u(\mathbf x) &= 1 \qquad\qquad & \text{in}\ \Omega, + \\ + u &= 0 \qquad\qquad & \text{on}\ \partial\Omega, +@f} +where $a(\mathbf x)$ is a spatially variable coefficient defined as +@f{align*}{ + a(\mathbf x) = + \begin{cases} + 20 & \text{if}\ |\mathbf x|<0.5, \\ + 1 & \text{otherwise.} + \end{cases} +@f} +

What adaptively refined meshes look like

-- 2.39.5