From: Wolfgang Bangerth Date: Wed, 11 Mar 2020 21:04:12 +0000 (-0600) Subject: Add links to step-20 and 22. X-Git-Tag: v9.2.0-rc1~428^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d5aa13a789b7c28f47a05e5d76ae6d2bdd240842;p=dealii.git Add links to step-20 and 22. --- diff --git a/examples/step-20/doc/intro.dox b/examples/step-20/doc/intro.dox index 49ebe13909..bda18d18e1 100644 --- a/examples/step-20/doc/intro.dox +++ b/examples/step-20/doc/intro.dox @@ -124,8 +124,12 @@ space $H({\textrm{div}})=\{{\mathbf w}\in L^2(\Omega)^d:\ {\textrm{div}}\ {\math for $\mathbf u$,$\mathbf v$, and $L^2$ for $p,q$. It is a well-known fact stated in almost every book on finite element theory that if one chooses discrete finite element spaces for the approximation of ${\mathbf u},p$ inappropriately, then the -resulting discrete saddle-point problem is instable and the discrete solution -will not converge to the exact solution. +resulting discrete problem is instable and the discrete solution +will not converge to the exact solution. (Some details on the problem +considered here -- which falls in the class of "saddle-point problems" +-- can be found on the Wikipedia page on the Ladyzhenskaya-Babuska-Brezzi +(LBB) condition.) To overcome this, a number of different finite element pairs for ${\mathbf u},p$ have been developed that lead to a stable discrete problem. One such pair is diff --git a/examples/step-22/doc/intro.dox b/examples/step-22/doc/intro.dox index f8f5664ffa..b62dd66ebf 100644 --- a/examples/step-22/doc/intro.dox +++ b/examples/step-22/doc/intro.dox @@ -351,11 +351,13 @@ for all test functions $\textbf{v}\in \textbf V_0 = \{\varphi \in H^1(\Omega)^d: \varphi_{\Gamma_D}=0\},q\in Q$. -These equations represent a symmetric saddle point problem. It is well known +These equations represent a symmetric saddle +point problem. It is well known that then a solution only exists if the function spaces in which we search for a solution have to satisfy certain conditions, typically referred to as the Babuska-Brezzi or Ladyzhenskaya-Babuska-Brezzi (LBB) conditions. The continuous -function spaces above satisfy them. However, when we discretize the equations by +function spaces above satisfy these. However, when we discretize the equations by replacing the continuous variables and test functions by finite element functions in finite dimensional spaces $\textbf V_{g,h}\subset \textbf V_g, Q_h\subset Q$, we have to make sure that $\textbf V_h,Q_h$ also satisfy the LBB