From 933bd77a8f2ff70cdb9882079166dd2e8a77e8e4 Mon Sep 17 00:00:00 2001 From: David Wells Date: Sat, 11 May 2019 14:09:23 -0400 Subject: [PATCH] step-38: delete trailing whitespace. --- examples/step-38/doc/intro.dox | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/examples/step-38/doc/intro.dox b/examples/step-38/doc/intro.dox index 440c18abc0..f28962a508 100644 --- a/examples/step-38/doc/intro.dox +++ b/examples/step-38/doc/intro.dox @@ -38,12 +38,12 @@ i.e. each point $\hat{\mathbf x}\in\hat S$ induces a point ${\mathbf G_S\dealcoloneq (D \mathbf{x}_S)^T \ D \mathbf{x}_S @f] denotes the corresponding first fundamental form, where $D -\mathbf{x}_S=\left(\frac{\partial x_{S,i}(\hat{\mathbf x})}{\partial \hat x_j}\right)_{ij}$ is the +\mathbf{x}_S=\left(\frac{\partial x_{S,i}(\hat{\mathbf x})}{\partial \hat x_j}\right)_{ij}$ is the derivative (Jacobian) of the mapping. In the following, $S$ will be either the entire surface $\Gamma$ or, more convenient for the finite element method, any face $S \in {\mathbb T}$, where ${\mathbb T}$ is a partition (triangulation) of $\Gamma$ -constituted of quadrilaterals. +constituted of quadrilaterals. We are now in position to define the tangential gradient of a function $v : S \rightarrow \mathbb R$ by @f[ @@ -56,7 +56,7 @@ Note that an alternate way to compute the surface gradient on smooth surfaces $\ \nabla_S v = \nabla \tilde v - \mathbf n (\mathbf n \cdot \nabla \tilde v), @f] where $\tilde v$ is a "smooth" extension of $v$ in a tubular neighborhood of $\Gamma$ and -$\mathbf n$ is the normal of $\Gamma$. +$\mathbf n$ is the normal of $\Gamma$. Since $\Delta_S = \nabla_S \cdot \nabla_S$, we deduce @f[ \Delta_S v = \Delta \tilde v - \mathbf n^T \ D^2 \tilde v \ \mathbf n - (\mathbf n \cdot \nabla \tilde v) (\nabla \cdot \mathbf n - \mathbf n^T \ D \mathbf n \ \mathbf n ). @@ -79,13 +79,13 @@ Moreover, each integral in the above expression is computed in the reference element $\hat K \dealcoloneq [0,1]^2$ so that @f{align*} -\int_{K} \nabla_{K} u \cdot \nabla_{K} v -&= +\int_{K} \nabla_{K} u \cdot \nabla_{K} v +&= \int_{\hat K} \nabla (u \circ \mathbf x_K)^T G_K^{-1} (D \mathbf x_K)^T D \mathbf x_K G_K^{-1} \nabla (v \circ \mathbf x_K) \sqrt{\det (G_K)} \\ -&= +&= \int_{\hat K} \nabla (u \circ \mathbf x_K)^T G_K^{-1} \nabla (v \circ \mathbf x_K) \sqrt{\det (G_K)} @f} @@ -126,7 +126,7 @@ return This provides exactly the terms we need for our computations. On a more general note, details for the finite element approximation on -surfaces can be found for instance in +surfaces can be found for instance in [Dziuk, in Partial differential equations and calculus of variations 1357, Lecture Notes in Math., 1988], [Demlow, SIAM J. Numer. Anal. 47(2), 2009] @@ -155,12 +155,12 @@ We produce one test case for a 2d problem and another one for 3d: is to project away the normal derivative as described above using the natural extension of $u(\mathbf x)$ (still denoted by $u$) over $\mathbb R^d$, i.e. to compute @f[ -\Delta_\Gamma u = \Delta u - \mathbf n^T \ D^2 u \ \mathbf n - (\mathbf n \cdot \nabla u)\ \kappa, - @f] + @f] where $\kappa$ is the total curvature of $\Gamma$. Since we are on the unit circle, $\mathbf n=\mathbf x$ and $\kappa = 1$ so that @f[ -\Delta_\Gamma u = -8 x_1x_2. - @f] + @f] A somewhat simpler way, at least for the current case of a curve in two-dimensional space, is to note that we can map the interval $t \in @@ -172,7 +172,7 @@ We produce one test case for a 2d problem and another one for 3d: segment of length $dt$ is mapped onto a piece of curve of exactly the same length, the tangential Laplacian then satisfies @f{align*} - \Delta_\Gamma u + \Delta_\Gamma u &= \frac{d^2}{dt^2}(-2\cos t \sin t) = -2 \frac{d}{dt}(-\sin^2 t + \cos^2 t) = -2 (-2 \sin t \cos t - 2 \cos t \sin t) @@ -191,13 +191,13 @@ We produce one test case for a 2d problem and another one for 3d: awkward and lengthy expression. You can find the full expression in the source code. - + In the program, we will also compute the $H^1$ seminorm error of the solution. Since the solution function and its numerical approximation are only defined on the manifold, the obvious definition of this error functional is -$| e |_{H^1(\Gamma)} - = | \nabla_\Gamma e |_{L_2(\Gamma)} +$| e |_{H^1(\Gamma)} + = | \nabla_\Gamma e |_{L_2(\Gamma)} = \left( \int_\Gamma | \nabla_\Gamma (u-u_h) |^2 \right)^{1/2}$. This requires us to provide the tangential gradient $\nabla_\Gamma u$ to the function VectorTools::integrate_difference (first introduced in step-7), which we -- 2.39.5