From: Krishnakumar Gopalakrishnan Date: Sun, 19 Jan 2020 20:07:31 +0000 (+0000) Subject: In step-7, removes the explicit listing of various norms since the list is anyway... X-Git-Tag: v9.2.0-rc1~574^2~1 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=8c78ee857458c78d48ef1f6f2cf82e36c36b69d6;p=dealii.git In step-7, removes the explicit listing of various norms since the list is anyway outdated --- diff --git a/examples/step-7/doc/intro.dox b/examples/step-7/doc/intro.dox index 57f1c13ec2..d54e667f9b 100644 --- a/examples/step-7/doc/intro.dox +++ b/examples/step-7/doc/intro.dox @@ -29,27 +29,8 @@ verification which is a very complicated problem. Rather we will demonstrate the tools which deal.II can offer in this respect. This is basically centered around the functionality of a single function, VectorTools::integrate_difference(). This function computes the difference between a given continuous function and -a finite element field in various norms on each cell. At the time of writing -this tutorial program, the norms this function can compute are the following, -where $u$ denotes the continuous function -and $u_h$ the finite element field, and $K$ is an element of the -triangulation: -@f{eqnarray*} - {\| u-u_h \|}_{L_1(K)} &=& \int_K |u-u_h| \; dx, - \\ - {\| u-u_h \|}_{L_2(K)} &=& \left( \int_K |u-u_h|^2 \; dx \right)^{1/2}, - \\ - {\| u-u_h \|}_{L_\infty(K)} &=& \max_{x \in K} |u(x) - u_h(x)|, - \\ - {| u-u_h |}_{H^1(K)} &=& \left( \int_K |\nabla(u-u_h)|^2 \; dx \right)^{1/2}, - \\ - {\| u-u_h \|}_{H^1(K)} &=& \left( {\| u-u_h \|}^2_{L_2(K)} - +{| u-u_h |}^2_{H^1(K)} \right)^{1/2}. -@f} -(All these norms and semi-norms can also be evaluated with weighting functions, -for example in order to exclude singularities from the determination of the -global error, and the function also works for vector-valued functions.) Of -course, like with any other integral, we can only evaluate these norms using quadrature formulas; +a finite element field in various norms on each cell. +Of course, like with any other integral, we can only evaluate these norms using quadrature formulas; the choice of the right quadrature formula is therefore crucial to the accurate evaluation of the error. This holds in particular for the $L_\infty$ norm, where we evaluate the maximal deviation of numerical and exact solution