From: JaeryunYim Date: Thu, 1 Sep 2016 10:38:21 +0000 (+0900) Subject: Minor edits. X-Git-Tag: v8.5.0-rc1~624^2~22 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=c2816adcada01fa4db64fc7d1394419f6e0bf08d;p=dealii.git Minor edits. --- diff --git a/include/deal.II/fe/fe_p1nc.h b/include/deal.II/fe/fe_p1nc.h index 8083ce6ce7..cb0b4857a0 100644 --- a/include/deal.II/fe/fe_p1nc.h +++ b/include/deal.II/fe/fe_p1nc.h @@ -30,37 +30,37 @@ DEAL_II_NAMESPACE_OPEN /** * Implementation for the scalar version of the P1 nonconforming finite - * element, a piecewise linear finite element on quadrilaterals in 2D. + * element, a piecewise linear element on quadrilaterals in 2D. * - * Unlike any continuous conforming finite element, - * it does not enforce the continuity across edges. - * But it requires the continuity just in weak sense: + * Unlike any continuous conforming finite element belongs to H^1_0, + * the P1 nonconforming element does not enforce the continuity across edges. + * But it requires the continuity just in integral 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. + * Thus each function in the nonconforming element space can be discontinuous, not included in H^1_0, as functions in Discontinuous Galerkin (DG) finite element spaces. + * Although any function in DG element space also has nonconformity, it is completely discontinuous across edges without any relation. + * It is a reason why usual weak formulations for DG schemes contain additional penalty terms for jump across edges to control discontinuity. + * However 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 function value at the mipoint of each edge is same to the mean value on the edge. + * Thus the continuity of the integral value across each edge is equivalent to + * the continuity of the midpoint value of each 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. + * The degrees of freedom (DOFs) on a quadrilateral are defined by midpoint values on edges. + * But these four DOFs 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. + * satisfies '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 of the another edge pair. * * \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, + * Conversely if 4 values at midpoints satisfying the dice rule are just 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, + * Due to the dice rule, three values at any three midpoints can 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. @@ -80,11 +80,11 @@ 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, - * and zero at two midpoints of other edges. - * Note that this situation satisfies the dice rule which is described above. - * We denote such a function by \phi_j. + * For each vertex v_j of given quad, there are two edges of which v_j is one of end points. + * Consider a linear function such that 0.5 value at two midpoints of such edges, + * and 0.0 at two midpoints of other edges. + * Note that the set of these values satisfies the dice rule which is described above. + * We denote such a function assoicated with vertex v_j by \phi_j. * The canonical (local) basis functions are given as any three shape functions of * the following four linear functions: @@ -157,17 +157,18 @@ DEAL_II_NAMESPACE_OPEN * 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, as the standard bilinear finite element @p Q_1. + * When a problem with homogeneous Dirichlet boundary condition is dealt, + * the total number of DOFs is equal to the number of interior nodes, as the standard bilinear finite element @p Q_1. *

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. + * Contrast with ordinary Lagrange finite elements, 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 (global) basis function associated with a node has 0.75 at that node, not 1.0. + * Thus we need an 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. + * It provides appropriate weights used in @p unit_support_points. + *

References

* You can find the paper about the P1NC element at * http://epubs.siam.org/doi/abs/10.1137/S0036142902404923.