From: Giovanni Alzetta Date: Wed, 2 May 2018 15:59:45 +0000 (+0200) Subject: Added description of C and a solution example X-Git-Tag: v9.1.0-rc1~1190^2~7 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=ddced554a89ab22b6233e9370750d3b51637a463;p=dealii.git Added description of C and a solution example --- diff --git a/examples/step-60/doc/intro.dox b/examples/step-60/doc/intro.dox index dd02773084..221f2b5c5a 100644 --- a/examples/step-60/doc/intro.dox +++ b/examples/step-60/doc/intro.dox @@ -201,6 +201,19 @@ C_{\alpha j} := (v_j, q_\alpha)_\Gamma = \sum_{K\in \Gamma} \int_{\hat K} \sum_{K\in \Gamma} \sum_{i=1}^{n_q} \big(\hat q_\alpha(\hat x_i) (v_j \circ F_{K}) (\hat x_i) J_K (\hat x_i) w_i \big) \f] +Computing this sum is difficult because we have to evaluate $(v_j \circ F_{K}) (\hat x_i)$; this +complicated process is illustrated in the following picture: + +

+ +

+ +This translates in the following algorithm: +- Fix the cell $K$ in $\Gamma$ and compute the real point $y_i \coloneqq F_{K} (\hat x_i)$ +- Find the cell of $\Omega$ in which $y_i$ lies, say it's $T$. +- To evaluate the basis function use invert shape function $G_{T}$ on $T$: + $v_j(y_i) = \hat v_j \circ G^{-1}_{T} (y_i)$. + We solve the above saddle point problem by iterating over the Schur complement (which is described, for example, in step-20), and we construct such complement using LinearOperator classes. diff --git a/examples/step-60/doc/results.dox b/examples/step-60/doc/results.dox index 85f416278d..c48f2a2d22 100644 --- a/examples/step-60/doc/results.dox +++ b/examples/step-60/doc/results.dox @@ -2,8 +2,20 @@

Test case 1:

-For the default problem the value of u on Gamma is 1 and on $\partial\Omega$ -is 0. In fact this is the solution: +For the default problem the value of u on Gamma is 1: this is like imposing a constant +Dirichlet boundary on the portion of $\Omega$ inside gamma. +

+ +

+In this second image we can appreciate how, using GridTools::compute_point_locations , +we have been able to adaptively refine $\Omega$ exactly where the solution is varying +the most: + +

+ +

Possibilities for extensions