From: Timo Heister Date: Mon, 23 Dec 2019 21:16:36 +0000 (-0500) Subject: small grammar fixes X-Git-Tag: v9.2.0-rc1~678^2~8 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=c55cea9686e4e330009f65a400cc7b154f06b8d5;p=dealii.git small grammar fixes --- diff --git a/examples/step-71/doc/intro.dox b/examples/step-71/doc/intro.dox index c29549ce6c..5f90e3d17a 100644 --- a/examples/step-71/doc/intro.dox +++ b/examples/step-71/doc/intro.dox @@ -108,7 +108,7 @@ does not make sense. Similar reasoning can be applied for 2d and 3d situations. In other words: This approach of trying to integrate over the entire -domain and the integrating by parts can't work. +domain and then integrating by parts can't work. Historically, numerical analysts have tried to address this by inventing finite elements that are "C1 continuous", i.e., that use @@ -118,8 +118,8 @@ element, the Clough-Tocher element and others, all developed in the late 1960s. From a twenty-first century perspective, they can only be described as bizarre in their construction. They are also exceedingly cumbersome to implement if one wants to use general meshes. As a -consequence, they have largely fallen out of favor and deal.II does -not contain implementations of these shape functions. +consequence, they have largely fallen out of favor and deal.II currently +does not contain implementations of these shape functions.

What to do instead?

@@ -209,7 +209,7 @@ for an equation that has two derivatives on each shape function. In analogy to the Interior Penalty (IP) method for the Laplace equation, this scheme for the biharmonic equation is typically called the $C^0$ IP (or C0IP) method, since it uses $C^0$ (continuous but not continuously -differentiable) shape functions with an interior penalty. +differentiable) shape functions with an interior penalty formulation.

Derivation of the $C^0$ IP method

@@ -217,7 +217,7 @@ differentiable) shape functions with an interior penalty. We base this program on the $C^0$ IP method presented by Susanne Brenner and Li-Yeng Sung in the paper "C$^0$ Interior Penalty Method for Linear Fourth Order Boundary Value Problems on polygonal -domains'' @cite Brenner2005 , where the method is +domains'' (@cite Brenner2005), where the method is derived for the biharmonic equation with "clamped" boundary conditions. @@ -343,7 +343,7 @@ stability of the method. We will discuss our choice in the program below. On polygonal domains, the weak solution $u$ to the biharmonic equation lives in $H^{2 +\alpha}(\Omega)$ where $\alpha \in(1/2, 2]$ is determined by the interior angles at the corners of $\Omega$. For -instance, whenever $\Omega$ is convex, $\alph=1$; $\alpha$ may be less +instance, whenever $\Omega$ is convex, $\alpha=1$; $\alpha$ may be less than one if the domain has re-entrant corners but $\alpha$ is close to $1$ if one of all interior angles is close to $\pi$. @@ -354,7 +354,8 @@ discretization outlined above yields the convergence rates as discussed below. -Convergence in the $C^0$ IP-norm: +Convergence in the $C^0$ IP-norm + Ideally, we would like to measure convergence in the "energy norm" $\|D^2(u-u_h)\|$. However, this does not work because, again, the discrete solution $u_h$ does not have two (weak) derivatives. Instead, @@ -387,23 +388,27 @@ that $\alpha\ge p-1$; this can only happen if (i) the domain is convex with a sufficiently smooth boundary, and (ii) $m\ge p-3$. In practice, of course, the solution is what it is (independent of the polynomial degree we choose), and the -last condition can then equivalent be read as saying that there is +last condition can then equivalently be read as saying that there is definitely no point in choosing $p$ large if $m$ is not also large. In other words, the only reasonably choices for $p$ are $p\le m+3$ because larger polynomial degrees do not result in higher convergence orders. -Convergence in the $L_2$-norm: The optimal convergence rate in -the $L_2$-norm is $\mathcal{O}(h^{p+1})$ provided $p \ge 3$. More -details can be found in Theorem 4.6 of @cite Engel2002 . +Convergence in the $L_2$-norm + +The optimal convergence rate in the $L_2$-norm is $\mathcal{O}(h^{p+1})$ +provided $p \ge 3$. More details can be found in Theorem 4.6 of @cite +Engel2002 . The default in the program below is to choose $p=2$. In that case, the theorem does not apply, and indeed one only gets $\mathcal{O}(h^2)$ instead of $\mathcal{O}(h^3)$ as we will show in the results section. -Convergence in the $H^1$-seminorm: Given that we expect +Convergence in the $H^1$-seminorm + +Given that we expect $\mathcal{O}(h^{p-1})$ in the best of cases for a norm equivalent to the $H^2$ seminorm, and $\mathcal{O}(h^{p+1})$ for the $L_2$ norm, one may ask about what happens in the $H^1$ seminorm that is intermediate diff --git a/examples/step-71/doc/results.dox b/examples/step-71/doc/results.dox index a9bb502d12..2501d2cdcb 100644 --- a/examples/step-71/doc/results.dox +++ b/examples/step-71/doc/results.dox @@ -154,10 +154,10 @@ $\gamma = p(p+1)$ is a better penalty. 5 2.822e-04 1.68 1.605e-03 1.90 2.041e-01 1.03 -In the case, all convergence rates of $u$ -follow the theoretical expectations. -But compared to the results from $\gamma = p(p+1)$, -it does not show a good convergence on $L_2$ errors. +In this case, all convergence rates of $u$ +follow the theoretical expectations, but, +compared to the results from $\gamma = p(p+1)$, +we don't see good convergence of the $L_2$ errors.

Conclusions for the choice of the penalty parameter