From d967f8bef6c608f0abe74f7f906042ad30d9a6ca Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Wed, 13 May 2020 07:37:08 -0600 Subject: [PATCH] Fix missing closing braces. --- examples/step-47/doc/results.dox | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/examples/step-47/doc/results.dox b/examples/step-47/doc/results.dox index a2b0ad71d9..088d096cc0 100644 --- a/examples/step-47/doc/results.dox +++ b/examples/step-47/doc/results.dox @@ -240,18 +240,18 @@ make sense: @f} and by the definition of jump over cell interfaces, @f{align*}{ - -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}). + -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}). @f} We separate interior faces and boundary faces of the domain, @f{align*}{ - -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}) - - \sum_{e \in \partial \Omega} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} h, + -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}) + - \sum_{e \in \partial \Omega} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}}} h, @f} where $\mathbb{F}^i$ is the set of interior faces. This leads us to @f{align*}{ - \sum_{K \in \mathbb{T}} \int_K (D^2 v_h) : (D^2 u_h) \ dx - \sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}) \ ds - = \sum_{K \in \mathbb{T}}\int_{K} v_h f \ dx + \sum_{e\subset\partial\Omega} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} h \ ds. + \sum_{K \in \mathbb{T}} \int_K (D^2 v_h) : (D^2 u_h) \ dx - \sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}) \ ds + = \sum_{K \in \mathbb{T}}\int_{K} v_h f \ dx + \sum_{e\subset\partial\Omega} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}}} h \ ds. @f} In order to symmetrize and stabilize the discrete problem, -- 2.39.5