From: Wolfgang Bangerth Date: Wed, 11 Mar 2020 21:18:43 +0000 (-0600) Subject: Clarify step-22. X-Git-Tag: v9.2.0-rc1~429^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F9652%2Fhead;p=dealii.git Clarify step-22. --- diff --git a/examples/step-22/doc/intro.dox b/examples/step-22/doc/intro.dox index f8f5664ffa..79b7af8051 100644 --- a/examples/step-22/doc/intro.dox +++ b/examples/step-22/doc/intro.dox @@ -118,8 +118,19 @@ over the domain $\Omega$, yielding the following set of equations: which has to hold for all test functions $\phi = \begin{pmatrix}\textbf{v} \\ q\end{pmatrix}$. -In practice, one wants to impose as little regularity on the pressure -variable as possible; consequently, we integrate by parts the second term: +A generally good rule of thumb is that if one can reduce how +many derivatives are taken on any variable in the formulation, then +one should in fact do that using integration by parts. (This is +motivated by the theory of partial +differential equations, and in particular the difference between +strong and weak +solutions.) We have already done that for the Laplace equation, +where we have integrated the second derivative by parts to obtain the +weak formulation that has only one derivative on both test and trial +function. + +In the current context, we integrate by parts the second term: @f{eqnarray*} (\textbf{v}, -2\; \textrm{div}\; \varepsilon(\textbf{u}))_{\Omega} - (\textrm{div}\; \textbf{v}, p)_{\Omega} @@ -149,6 +160,10 @@ defined as 2 \int_\Omega \sum_{i,j=1}^d \frac{\partial v_j}{\partial x_i} \varepsilon(\textbf{u})_{ij} \ dx. @f} +Using this, we have now reduced the requirements on our variables to +first derivatives for $\mathbf u,\mathbf v$ and no derivatives at all +for $p,q$. + Because the scalar product between a general tensor like $\nabla\textbf{v}$ and a symmetric tensor like $\varepsilon(\textbf{u})$ equals the scalar product between the