From 11d9c66a88e1f2549f5ac4aa30b3cf0e416260be Mon Sep 17 00:00:00 2001 From: bonito Date: Wed, 5 Jan 2011 20:56:29 +0000 Subject: [PATCH] git-svn-id: https://svn.dealii.org/trunk@23129 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-38/doc/intro.dox | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index ebdcdf889a..034de548e2 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -41,7 +41,7 @@ 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)\circ \mathbf x_S := D \mathbf x_S \ G_S^{-1} \ D (v \circ \mathbf x_S). +(\nabla_S v)\circ \mathbf x_S := D \mathbf x_S \ G_S^{-1} \ \nabla (v \circ \mathbf x_S). @f] The surface Laplacian (also called the Laplace-Beltrami operator) is then defined as $\Delta_S:= \nabla_S \cdot \nabla_S$. @@ -66,7 +66,7 @@ solutions). We therefore resort to the weak formulation @f] and take advantage of the partition ${\mathbb T}$ to further write @f[ -\sum_{K\in {\mathbb T}}\int_K \nabla_K u \cdot \nabla_K v = \sum_{K\in +\sum_{K\in {\mathbb T}}\int_K \nabla_{K} u \cdot \nabla_{K} v = \sum_{K\in {\mathbb T}} \int_K f \ v \qquad \forall v \in H^1_0(\Gamma). @f] Moreover, each integral in the above expression is computed in the reference @@ -75,12 +75,12 @@ so that @f{align*} &\int_{K} \nabla_{K} u \cdot \nabla_{K} v \\ &= -\int_{\hat K} D(u \circ \mathbf x_K)^T G_K^{-1} (D \mathbf - x_K)^T D \mathbf x_K G_K^{-1} D(v \circ \mathbf x_K) \sqrt{\det +\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} D(u \circ \mathbf x_K)^T G_K^{-1} D(v \circ \mathbf x_K) \sqrt{\det +\int_{\hat K} \nabla (u \circ \mathbf x_K)^T G_K^{-1} \nabla (v \circ \mathbf x_K) \sqrt{\det (G_K)} @f} and @@ -94,7 +94,7 @@ Finally, we use a quadrature formula defined by points $\{p_l\}_{l=1}^N\subset evaluate the above integrals and obtain @f[\int_{K} \nabla_{K} u \cdot \nabla_{K} v \approx \sum_{l=1}^N - (D(u \circ \mathbf x_K)(p_l))^T G^{-1}(p_l) D(v \circ \mathbf x_K) + (\nabla (u \circ \mathbf x_K)(p_l))^T G^{-1}(p_l) \nabla (v \circ \mathbf x_K) (p_l) \sqrt{\det (G(p_l))} \ w_l @f] and -- 2.39.5