From: JaeryunYim Date: Thu, 1 Sep 2016 07:48:28 +0000 (+0900) Subject: Add X-Git-Tag: v8.5.0-rc1~624^2~23 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=25c766dab6cb233f172aa2751f637bda5b64b98f;p=dealii.git Add --- diff --git a/include/deal.II/fe/fe_p1nc.h b/include/deal.II/fe/fe_p1nc.h index 61def02800..8083ce6ce7 100644 --- a/include/deal.II/fe/fe_p1nc.h +++ b/include/deal.II/fe/fe_p1nc.h @@ -33,13 +33,20 @@ DEAL_II_NAMESPACE_OPEN * element, a piecewise linear finite element on quadrilaterals in 2D. * * 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. + * it does not enforce the continuity across edges. + * But it requires the continuity just in weak sense: + * any function in the space should have the same integral values on two sides of the common edge shared by two adjacent elements. + + * Thus any function in the nonconforming element spaces is discontinuous, not included in H^1_0, as the case of DG finite elements. + * Although any function in DG finite element space has nonconformity also, it is completely discontinuous across edges. + * It is the reason why usual weak formulations for DG schemes contain additional penalty terms for jump across edges to stabilize. + * However the nonconforming elements usually do not need additional terms in their weak formulations due to the continuity in integral on edges. + + *

DOFs and Dice Rule

+ * Since any function in the P1 nonconforming space is piecewise linear on each element, + * the function value at the mipoint of the edge is same to the mean value on the edge. + * Thus the continuity of the integral value across the edge is equivalent to + * the continuity of the midpoint value of the edge in this case. * * 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. @@ -49,13 +56,17 @@ DEAL_II_NAMESPACE_OPEN * * \phi(m_0) + \phi(m_1) = \phi(m_2) + \phi(m_3). * + * Conversely if just 4 values at midpoints satisfying the dice rule are given, + * then there always exists the unique linear function which coincides with 4 midpoints values. + * * 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. - * Shape functions + *

Shape functions

+ * @verbatim * 2---------|---------3 * | | * | | @@ -67,6 +78,7 @@ DEAL_II_NAMESPACE_OPEN * | | * | | * 0---------|---------1 + * @endverbatim * 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, @@ -77,8 +89,9 @@ DEAL_II_NAMESPACE_OPEN * The canonical (local) basis functions are given as any three shape functions of * the following four linear functions: - * shape function \phi_0 - + * - * Note that this shape functions are constructed on each cell, not on the reference cell only. - * get_linear_shape computes the coefficients for shape functions when fill_fe_values is called on each cell. + * Note that above shape functions are constructed on each cell, not on the reference cell only. + * @p get_linear_shape computes the coefficients for shape functions when @p fill_fe_values is called on each cell. * The (global) basis function associated with a node is defined by the composition of * (local) basis functions associated with the node on each element. * In case of the problem with homogeneous Dirichlet boundary condition, - * the number of DOFs is equal to the number of interior nodes. + * the number of DOFs is equal to the number of interior nodes, as the standard bilinear finite element @p Q_1. - * (TODO: unit_support_points) + *

Unit support points

+ * Contrast with ordinary Lagrange finite elements, the DOF value with respect to the P1 nonconforming element at given node does not coincide with the function value at that node. + * For instance, the shape function \phi_0 which is associated with vertex 0 has 0.75 at vertex 0, not 1.0. + * Thus we need a interpolation operator which maps any smooth function into a function with proper DOF values in the P1 element space. + * One natural interpolant associated with given smooth function is the linear function whose midpoint value at each edge is defined by + * the average of two values at endpoints of the edge. + * In other word, it provides appropriate weights used in @p unit_support_points. * 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> @@ -286,7 +307,7 @@ private: -/** @}*/ +/*@}*/ DEAL_II_NAMESPACE_CLOSE