From 85a0544cd056adb8bee07215938b592daca79083 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Mon, 30 Jan 2017 23:02:29 +0100 Subject: [PATCH] Fix broken equations in glossary entry on generalized support points. --- doc/doxygen/headers/glossary.h | 32 ++++++++++++++++---------------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/doc/doxygen/headers/glossary.h b/doc/doxygen/headers/glossary.h index d9ade721ab..5b75cecdd7 100644 --- a/doc/doxygen/headers/glossary.h +++ b/doc/doxygen/headers/glossary.h @@ -589,7 +589,7 @@ * used in the finite element community to indicate two slightly different, * but related things. The first is that we'd like to represent the finite * element solution as a linear combination of shape functions, in the form - * $u_h(\mathbf x) = \sum_{j=0}^{N-1} U_j \varphi_j(\mathbf x)$. Here, $U_j$ + * $u_h(\mathbf{x}) = \sum_{j=0}^{N-1} U_j \varphi_j(\mathbf{x})$. Here, $U_j$ * is a vector of expansion coefficients. Because we don't know their values * yet (we will compute them as the solution of a linear or nonlinear system), * they are called "unknowns" or "degrees of freedom". The second meaning of @@ -600,7 +600,7 @@ * V_h$). In other words, all we say here that the solution needs to lie in * some space $V_h$. However, to actually solve this problem on a computer we * need to choose a basis of this space; this is the set of shape functions - * $\varphi_j(\mathbf x)$ we have used above in the expansion of $u_h(\mathbf + * $\varphi_j(\mathbf{x})$ we have used above in the expansion of $u_h(\mathbf * x)$ with coefficients $U_j$. There are of course many bases of the space * $V_h$, but we will specifically choose the one that is described by the * finite element functions that are traditionally defined locally on the @@ -859,19 +859,19 @@ * generalization of @ref GlossSupport "support points". The latter * are used to describe that a finite element simply interpolates * values at individual points (the "support points"). If we call these - * points $\hat\mathbf x_i$ (where the hat indicates that these points - * are defined on the reference cell $\hat K$), then one typically defines - * shape functions $\varphi_j(\mathbf x)$ in such a way that the + * points $\hat{\mathbf{x}}_i$ (where the hat indicates that these points + * are defined on the reference cell $\hat{K}$), then one typically defines + * shape functions $\varphi_j(\mathbf{x})$ in such a way that the * nodal functionals $\Psi_i[\cdot]$ simply evaluate the function - * at the support point, i.e., that $\Psi_i[\varphi]=\varphi(\hat\mathbf x_i)$, + * at the support point, i.e., that $\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)$, * and the basis is chosen so that $\Psi_i[\varphi_j]=\delta_{ij}$ where * $\delta_{ij}$ is the Kronecker delta function. This leads to the common * @ref GlossLagrange "Lagrange elements". * * (In the vector valued case, the only other piece of information - * besides the support points $\hat\mathbf x_i$ that one needs to provide + * besides the support points $\hat{\mathbf{x}}_i$ that one needs to provide * is the vector component $c(i)$ the $i$th node functional - * corresponds, so that $\Psi_i[\varphi]=\varphi(\hat\mathbf x_i)_{c(i)}$.) + * corresponds, so that $\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)_{c(i)}$.) * * On the other hand, there are other kinds of elements that are not * defined this way. For example, for the lowest order Raviart-Thomas element @@ -880,17 +880,17 @@ * components, but the normal component of this vector: * $\Psi_i[\varphi] * = - * \varphi(\hat\mathbf x_i) \cdot \mathbf n_i - * $, where the $\mathbf n_i$ are the normal vectors to the face of the cell - * on which $\hat\mathbf x_i$ is located. In other words, the node functional + * \varphi(\hat{\mathbf{x}}_i) \cdot \mathbf{n}_i + * $, where the $\mathbf{n}_i$ are the normal vectors to the face of the cell + * on which $\hat{\mathbf{x}}_i$ is located. In other words, the node functional * is a linear combination of the components of $\varphi$ when - * evaluated at $\hat\mathbf x_i$. Similar things happen for the BDM, + * evaluated at $\hat{\mathbf{x}}_i$. Similar things happen for the BDM, * ABF, and Nedelec elements (see the FE_BDM, FE_ABF, FE_Nedelec classes). * * In these cases, the element does not have support points because * it is not purely interpolatory; however, some kind of interpolation * is still involved when defining shape functions as the node functionals - * still require point evaluations at special points $\hat\mathbf x_i$. + * still require point evaluations at special points $\hat{\mathbf{x}}_i$. * In these cases, we call the points generalized support points. * * Finally, there are elements that still do not fit into this @@ -900,9 +900,9 @@ * functions, * $\Psi_i[\varphi] * = - * \int_{\hat K} \varphi(\hat\mathbf x) - * {\hat x_1}^{p_1(i)} - * {\hat x_2}^{p_2(i)} + * \int_{\hat{K}} \varphi(\hat{\mathbf{x}}) + * {\hat{x}_1}^{p_1(i)} + * {\hat{x}_2}^{p_2(i)} * $ in 2d, and similarly for 3d, where the $p_d(i)$ are the order * of the moment described by shape function $i$. Some other elements * use moments over edges or faces. In all of these cases, node functionals -- 2.39.5