From: Wolfgang Bangerth Date: Fri, 20 Oct 2017 21:51:19 +0000 (-0600) Subject: Rearrange terms in the formula. X-Git-Tag: v9.0.0-rc1~905^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=8c83fac2171229772dd9c170f05106b78f2bc89a;p=dealii.git Rearrange terms in the formula. --- diff --git a/examples/step-9/doc/intro.dox b/examples/step-9/doc/intro.dox index ddce0318a9..14d1e46a9f 100644 --- a/examples/step-9/doc/intro.dox +++ b/examples/step-9/doc/intro.dox @@ -121,11 +121,11 @@ for all discrete test functions $v_h$ there holds @f[ (\beta \cdot \nabla u_h, v_h + \delta \beta\cdot\nabla v_h)_\Omega - - (\beta\cdot {\mathbf n} u_h, v_h)_{\partial\Omega_-} + (u_h, \beta\cdot {\mathbf n} v_h)_{\partial\Omega_-} = (f, v_h + \delta \beta\cdot\nabla v_h)_\Omega - - (\beta\cdot {\mathbf n} g, v_h)_{\partial\Omega_-}. + (g, \beta\cdot {\mathbf n} v_h)_{\partial\Omega_-}. @f] @@ -136,18 +136,18 @@ to be inverted of the form (\beta \cdot \nabla \varphi_i, \varphi_j + \delta \beta\cdot\nabla \varphi_j)_\Omega - - (\beta\cdot {\mathbf n} \varphi_i, \varphi_j)_{\partial\Omega_-}, + (\varphi_i, \beta\cdot {\mathbf n} \varphi_j)_{\partial\Omega_-}, @f] with basis functions $\varphi_i,\varphi_j$. However, this is a pitfall that happens to every numerical analyst at least once (including the author): we have here expanded the solution -$u_h = \sum_i u_i \varphi_i$, but if we do so, we will have to solve the +$u_h = \sum_i U_i \varphi_i$, but if we do so, we will have to solve the problem @f[ - {\mathbf u}^T A = {\mathbf f}^T, + U^T A = F^T, @f] -where ${\mathbf u}=(u_i)$, i.e., we have to solve the transpose problem of -what we might have expected naively. +where $U$ is the vector of expansion coefficients, i.e., we have to +solve the transpose problem of what we might have expected naively. This is a point we made in the introduction of step-3. There, we argued that to avoid this very kind of problem, one should get in the habit of always