From: Nicola Giuliani Date: Thu, 9 Apr 2015 13:54:53 +0000 (+0200) Subject: improved the documentation of QTelles X-Git-Tag: v8.3.0-rc1~305^2~1 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=fdcda03997f96ee7442bdedc0889b3a5f93f957e;p=dealii.git improved the documentation of QTelles --- diff --git a/include/deal.II/base/quadrature_lib.h b/include/deal.II/base/quadrature_lib.h index 0400b13eac..b0ba4d865e 100644 --- a/include/deal.II/base/quadrature_lib.h +++ b/include/deal.II/base/quadrature_lib.h @@ -457,21 +457,21 @@ public: * International Journal for Numerical Methods in Engineering, vol 24, * pages 959–973. year 1987, the author applies the transformation on the * reference cell $[-1, 1]$ getting - * @f{align*} n(1) &= 1, \\ n(-1) &= -1, \\ dn/dx $= 0 at x = x_0, \\ - d2n/dx2 = 0 at x &= x_0 @f} + \begin{align*} n(1) &= 1, \\ n(-1) &= -1, \\ \frac{dn}{dx} &= 0 \text{ at } + x = x_0, \\ \frac{d^2n}{dx^2} &= 0 \text{ at } x = x_0 \end{align*} * We get - * @f{align*} a &= 1/q, \\ b &= -3gamma_bar/q, \\ c &= 3gamma_bar/q, \\ - d &= -b, @f} + \begin{align*} a &= \frac{1}{q}, \\ b &= -3 \frac{\bar{\Gamma}}{q}, \\ + c &= 3\frac{\bar{\Gamma}}{q}, \\ d &= -b, \end{align*} * with - * @f{align*} eta_star &= eta_bar^2 - 1, \\ gamma_bar 6 &= nthroot( eta_bar - * eta_star + abs(eta_star) ,3) + nthroot(eta_bar*eta_star - abs(eta_star),3) - + eta_bar, \\ q &= ((gamma-gamma_bar).^3 + gamma_bar*(gamma_bar^2+3)) - /(1+3*gamma_bar^2) @f}. - * Since the library assumes [0,1] as reference interval, we will map + \begin{align*} \eta^{*} &= \bar{\eta}^2 - 1, \\ \bar{\Gamma} &= \sqrt[3]{ + \bar{\eta} \eta^{*} + |\eta^{*} | } + \sqrt[3]{ \bar{\eta} \eta^{*} - + |\eta^{*} | } + \bar{\eta}, \\ q &= (\Gamma-\bar{\Gamma})^3 + \bar{\Gamma} + \frac{\bar{\Gamma}^2+3}{1+3\bar{\Gamma}^2} \end{align*} + * Since the library assumes $[0,1]$ as reference interval, we will map * these values on the proper reference interval in the implementation. * * This variable change can be used to integrate singular integrals. - * One example is $\f(x)/abs(x-x_0)$ on the reference interval $[0,1]$, + * One example is $f(x)/|x-x_0|$ on the reference interval $[0,1]$, * where $x_0$ is given at construction time, and is the location of the * singularity $x_0$, and $f(x)$ is a smooth non singular function. *