From 4359aea79f34aa96fcb46462bfecdb2a79a7bd1c Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Tue, 3 Jul 2018 09:59:37 -0600 Subject: [PATCH] Update the documentation of the Rannacher Turek element. --- include/deal.II/fe/fe_rannacher_turek.h | 46 +++++++++++++++++++------ 1 file changed, 35 insertions(+), 11 deletions(-) diff --git a/include/deal.II/fe/fe_rannacher_turek.h b/include/deal.II/fe/fe_rannacher_turek.h index c862bd5bc2..0522a280f9 100644 --- a/include/deal.II/fe/fe_rannacher_turek.h +++ b/include/deal.II/fe/fe_rannacher_turek.h @@ -29,24 +29,48 @@ DEAL_II_NAMESPACE_OPEN /** - * Implementation of Rannacher-Turek elements. Functions generated by this - * element will be discontinuous, but their jump along faces is mean value - * free. + * Implementation of the Rannacher-Turek element. This element is used + * to generate a stable pair of function spaces for the Stokes + * equation without having to increase the polynomial degree of the + * velocity space as much as one would do for the stable Taylor-Hood + * element which uses the $Q_2^d\times Q_1$ pair for velocity and + * pressure. That said, like many other non-conforming elements, it + * can also be used for the discretization of the Laplace + * equation. The element was first described in R. Rannacher and + * S. Turek: "Simple non-conforming quadrilateral Stokes element", + * Numerical Methods for Partial Differential Equations, vol. 8, + * pp. 97-112, 1992. * - * Implemented only in dimension 2, lowest order, without hanging nodes and + * The shape functions generated by this element are in general + * discontinuous, and consequently the element is not $H^1$ conforming + * (i.e., it is a "non-conforming" element). However, the shape + * functions are constructed in such a way that the jump along faces + * has mean value zero, and consequently there is some sort of + * conformity in the element: a conforming element would have a + * pointwise zero jump, a completely discontinuous element like the + * FE_DGQ elements can have entirely arbitrary values for the jump + * across a face, and the current element is somewhere in the middle + * because its jump is nonzero but at least has mean value zero. + * + * The element is currently implemented only in dimension 2, for the + * lowest polynomial order, and without hanging nodes and * restriction/prolongation. * + * *

Interpolation

* - *

Node values

The - * @ref GlossNodes "node values" - * are moments on faces. + *

Node values

+ * + * The @ref GlossNodes "node values" are moments on faces. + * + *

Generalized support points

* - *

Generalized support points

To calculate the node values, we are - * using a QGauss rule on each face. By default, we are using a two point rule - * to integrate Rannacher-Turek functions exactly. But in order to be able to + * To calculate the node values, we are using a QGauss rule on each + * face. By default, we are using a two point rule to integrate + * Rannacher-Turek functions exactly. But in order to be able to * interpolate other functions with sufficient accuracy, the number of - * quadrature points used on a face can be adjusted in the constructor. + * quadrature points used on a face can be adjusted in the + * constructor. * * @ingroup fe * @author Patrick Esser -- 2.39.5