From: Wolfgang Bangerth Date: Mon, 29 Nov 2010 15:30:26 +0000 (+0000) Subject: Add a note on the difference between using the symmetric gradient or the X-Git-Tag: v8.0.0~4785 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=71f10511651d690e880ea84f31633e38429360d8;p=dealii.git Add a note on the difference between using the symmetric gradient or the straight one. git-svn-id: https://svn.dealii.org/trunk@22874 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index 93ff3a273d..7ab2d55c99 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -47,6 +47,22 @@ become clear once we discuss the weak form of the equations. The equations covered here fall into the class of vector-valued problems. A toplevel overview of this topic can be found in the @ref vector_valued module. +@note Different people mean different things when they talk about the Stokes +equations. We here adopt the convention of the solid mechanics community which +uses the symmetric gradient. This differs from the convention in the fluids +community by a factor of two since the fact that $\textrm{div}\; \textbf{u}=0$ +implies that $-\textrm{div}\; \varepsilon(\textbf{u}) = \frac 12 \Delta +\textbf{u}$. The equations above are therefore equivalent to +@f{eqnarray*} + -\frac 12 \Delta\textbf{u} + \nabla p &=& \textbf{f}, + \\ + -\textrm{div}\; \textbf{u} &=& 0. +@f} +The Stokes equations that the fluid dynamics community would refer to do not +have the factor $\frac 12$ in front. Depending on your viewpoint you may +therefore want to adjust the bilinear form in the program when building the +system matrix and preconditioner. +

Weak form