From cc32f09eb33593cd7365796bcf3782ffe2e8e74e Mon Sep 17 00:00:00 2001 From: JaeryunYim Date: Thu, 4 Aug 2016 17:22:13 +0900 Subject: [PATCH] Add the first description of FE_P1NC --- include/deal.II/fe/fe_p1nc.h | 52 ++++++++++++++++++++++++++++++++++++ 1 file changed, 52 insertions(+) diff --git a/include/deal.II/fe/fe_p1nc.h b/include/deal.II/fe/fe_p1nc.h index f9283b4d5c..b5edb00590 100644 --- a/include/deal.II/fe/fe_p1nc.h +++ b/include/deal.II/fe/fe_p1nc.h @@ -28,6 +28,58 @@ DEAL_II_NAMESPACE_OPEN /*!@addtogroup fe */ /*@{*/ +/** + * Implementation for the scalar version of the P1 nonconforming finite + * element, a piecewise linear finite element on quadrilaterals. + * + * Unlike any continuous conforming finite element, + * it does not have the continuity across edges. + * But it requires the continuity in weak sense: + * a function in the space should have the same integral values on two sides of the common edge shared by two adjacent elements. + * + * Since any function in the space is piecewise linear on each element, + * the continuity of the integral value across the edge is equivalent to + * the continuity of the value at the midpoint of the edge. + * + * The degrees of freedom on a quadrilateral are given by midpoint values on edges. + * However these four dofs in 2D are not independent in fact. + * The simple observation reads that any linear function on a quadrilateral + * satisfies 'the dice rule': the sum of two function values at two midpoints of the edge pair on opposite + * position is equal to the sum of those on the another edge pair. + * + * \phi(m_0) + \phi(m_1) = \phi(m_2) + \phi(m_3). + * + * Due to the dice rule, three values at any three midpoints determine the last value at the last midpoint. + * It means that the genuine number of independent dofs on a quad is 3, + * and it is the same number to the dimension of the linear polynomial space in 2D. + * + * The canonical basis functions are given as any three shape functions of + * the following four linear functions: + * + * \phi_1, \phi_2, \phi_3, \phi_4. + * + * for each vertex v_j, there are two edges of which v_j is one of the end points. + * Consider the linear function such that one half at two midpoints of such edges, + * and zero at two midpoints of other edges. + + + * 2 -------|------- 3 + * | | + * | | + * | | + * | | + * - - + * | | + * | | + * | | + * | | + * 0 -------|------- 1 + + + * You can find the paper about the P1NC element at + * http://epubs.siam.org/doi/abs/10.1137/S0036142902404923. + + **/ class FE_P1NC : public FiniteElement<2,2> { -- 2.39.5