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[
\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 ).
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}
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]
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
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)
awkward and lengthy expression. You can find the full expression in the
source code.
</li>
-</ul>
+</ul>
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
<i>tangential</i> gradient $\nabla_\Gamma u$ to the function VectorTools::integrate_difference
(first introduced in step-7), which we
projects / dealii.git / commitdiff