From 0b510eaa57bb73f6301b94efb99302244526dd96 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Thu, 9 May 2019 21:11:48 -0600 Subject: [PATCH] Add some more information on the BR element. --- include/deal.II/fe/fe_bernardi_raugel.h | 30 +++++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/include/deal.II/fe/fe_bernardi_raugel.h b/include/deal.II/fe/fe_bernardi_raugel.h index 9f54989f87..bfd04c314c 100644 --- a/include/deal.II/fe/fe_bernardi_raugel.h +++ b/include/deal.II/fe/fe_bernardi_raugel.h @@ -33,7 +33,37 @@ DEAL_II_NAMESPACE_OPEN /** * The Bernardi-Raugel element. * + * This class implements the non-standard Bernardi-Raugel (BR) element + * that can be used as one part of a stable velocity/pressure pair for + * the Stokes equation. The BR element can be seen as either an + * enriched version of the $Q_1^d$ element with added bubble functions + * on each edge (in 2d) or face (in 3d), or as a reduced version of + * the $Q_2^d$ element. It addresses the fact that the $Q_1^d\times + * Q_0$ combination is not inf-sup stable (requiring a larger velocity + * space), and that the $Q_2^d\times Q_1$ combination is stable but + * sub-optimal since the velocity space is too large relative to the + * pressure space to provide additional accuracy commensurate with the + * cost of the large number of velocity unknowns. + * + * The element was introduced in the following paper: + * @code{.bib} + * @article{BR85, + * author = {Christine Bernardi and Genevi{\`e}ve Raugel}, + * title = {Analysis of some finite elements for the {S}tokes problem}, + * journal = {Mathematics of Computation}, + * publisher = {American Mathematical Society ({AMS})}, + * volume = {44}, + * number = {169}, + * pages = {71--79}, + * year = {1985}, + * doi = {10.1090/s0025-5718-1985-0771031-7}, + * url = {https://doi.org/10.1090/s0025-5718-1985-0771031-7} + * } + * @endcode + * + * *

Degrees of freedom

+ * * The BR1 element has dim degrees of freedom on each vertex and 1 on * each face. The shape functions are ordered by the $(Q_1)^d$ shape functions * supported on each vertex, increasing according to vertex ordering on the -- 2.39.5