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//
// Copyright (C) 2000 - 2015 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__fe_q_h
#define dealii__fe_q_h
#include <deal.II/base/config.h>
#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/fe/fe_q_base.h>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup fe */
/*@{*/
/**
* Implementation of a scalar Lagrange finite element @p Qp that yields the
* finite element space of continuous, piecewise polynomials of degree @p p in
* each coordinate direction. This class is realized using tensor product
* polynomials based on equidistant or given support points.
*
* The standard constructor of this class takes the degree @p p of this finite
* element. Alternatively, it can take a quadrature formula @p points defining
* the support points of the Lagrange interpolation in one coordinate
* direction.
*
* For more information about the <tt>spacedim</tt> template parameter check
* the documentation of FiniteElement or the one of Triangulation.
*
* <h3>Implementation</h3>
*
* The constructor creates a TensorProductPolynomials object that includes the
* tensor product of @p LagrangeEquidistant polynomials of degree @p p. This
* @p TensorProductPolynomials object provides all values and derivatives of
* the shape functions. In case a quadrature rule is given, the constructor
* creates a TensorProductPolynomials object that includes the tensor product
* of @p Lagrange polynomials with the support points from @p points.
*
* Furthermore the constructor fills the @p interface_constraints, the @p
* prolongation (embedding) and the @p restriction matrices. These are
* implemented only up to a certain degree and may not be available for very
* high polynomial degree.
*
*
* <h3>Numbering of the degrees of freedom (DoFs)</h3>
*
* The original ordering of the shape functions represented by the
* TensorProductPolynomials is a tensor product numbering. However, the shape
* functions on a cell are renumbered beginning with the shape functions whose
* support points are at the vertices, then on the line, on the quads, and
* finally (for 3d) on the hexes. To be explicit, these numberings are listed
* in the following:
*
* <h4>Q1 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0-------1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2-------3
* | |
* | |
* | |
* 0-------1
* @endverbatim
*
* <li> 3D case:
* @verbatim
* 6-------7 6-------7
* /| | / /|
* / | | / / |
* / | | / / |
* 4 | | 4-------5 |
* | 2-------3 | | 3
* | / / | | /
* | / / | | /
* |/ / | |/
* 0-------1 0-------1
* @endverbatim
*
* The respective coordinate values of the support points of the shape
* functions are as follows:
* <ul>
* <li> Shape function 0: <tt>[0, 0, 0]</tt>;
* <li> Shape function 1: <tt>[1, 0, 0]</tt>;
* <li> Shape function 2: <tt>[0, 1, 0]</tt>;
* <li> Shape function 3: <tt>[1, 1, 0]</tt>;
* <li> Shape function 4: <tt>[0, 0, 1]</tt>;
* <li> Shape function 5: <tt>[1, 0, 1]</tt>;
* <li> Shape function 6: <tt>[0, 1, 1]</tt>;
* <li> Shape function 7: <tt>[1, 1, 1]</tt>;
* </ul>
* </ul>
*
* In 2d, these shape functions look as follows: <table> <tr> <td
* align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q1/Q1_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q1/Q1_shape0001.png
* </td> </tr> <tr> <td align="center"> $Q_1$ element, shape function 0 </td>
*
* <td align="center"> $Q_1$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q1/Q1_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q1/Q1_shape0003.png
* </td> </tr> <tr> <td align="center"> $Q_1$ element, shape function 2 </td>
*
* <td align="center"> $Q_1$ element, shape function 3 </td> </tr> </table>
*
*
* <h4>Q2 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0---2---1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2---7---3
* | |
* 4 8 5
* | |
* 0---6---1
* @endverbatim
*
* <li> 3D case:
* @verbatim
* 6--15---7 6--15---7
* /| | / /|
* 12 | 19 12 1319
* / 18 | / / |
* 4 | | 4---14--5 |
* | 2---11--3 | | 3
* | / / | 17 /
* 16 8 9 16 | 9
* |/ / | |/
* 0---10--1 0---10--1
*
* *-------* *-------*
* /| | / /|
* / | 23 | / 25 / |
* / | | / / |
* * | | *-------* |
* |20 *-------* | |21 *
* | / / | 22 | /
* | / 24 / | | /
* |/ / | |/
* *-------* *-------*
* @endverbatim
* The center vertex has number 26.
*
* The respective coordinate values of the support points of the shape
* functions are as follows:
* <ul>
* <li> Shape function 0: <tt>[0, 0, 0]</tt>;
* <li> Shape function 1: <tt>[1, 0, 0]</tt>;
* <li> Shape function 2: <tt>[0, 1, 0]</tt>;
* <li> Shape function 3: <tt>[1, 1, 0]</tt>;
* <li> Shape function 4: <tt>[0, 0, 1]</tt>;
* <li> Shape function 5: <tt>[1, 0, 1]</tt>;
* <li> Shape function 6: <tt>[0, 1, 1]</tt>;
* <li> Shape function 7: <tt>[1, 1, 1]</tt>;
* <li> Shape function 8: <tt>[0, 1/2, 0]</tt>;
* <li> Shape function 9: <tt>[1, 1/2, 0]</tt>;
* <li> Shape function 10: <tt>[1/2, 0, 0]</tt>;
* <li> Shape function 11: <tt>[1/2, 1, 0]</tt>;
* <li> Shape function 12: <tt>[0, 1/2, 1]</tt>;
* <li> Shape function 13: <tt>[1, 1/2, 1]</tt>;
* <li> Shape function 14: <tt>[1/2, 0, 1]</tt>;
* <li> Shape function 15: <tt>[1/2, 1, 1]</tt>;
* <li> Shape function 16: <tt>[0, 0, 1/2]</tt>;
* <li> Shape function 17: <tt>[1, 0, 1/2]</tt>;
* <li> Shape function 18: <tt>[0, 1, 1/2]</tt>;
* <li> Shape function 19: <tt>[1, 1, 1/2]</tt>;
* <li> Shape function 20: <tt>[0, 1/2, 1/2]</tt>;
* <li> Shape function 21: <tt>[1, 1/2, 1/2]</tt>;
* <li> Shape function 22: <tt>[1/2, 0, 1/2]</tt>;
* <li> Shape function 23: <tt>[1/2, 1, 1/2]</tt>;
* <li> Shape function 24: <tt>[1/2, 1/2, 0]</tt>;
* <li> Shape function 25: <tt>[1/2, 1/2, 1]</tt>;
* <li> Shape function 26: <tt>[1/2, 1/2, 1/2]</tt>;
* </ul>
* </ul>
*
*
* In 2d, these shape functions look as follows (the black plane corresponds
* to zero; negative shape function values may not be visible): <table> <tr>
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0001.png
* </td> </tr> <tr> <td align="center"> $Q_2$ element, shape function 0 </td>
*
* <td align="center"> $Q_2$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0003.png
* </td> </tr> <tr> <td align="center"> $Q_2$ element, shape function 2 </td>
*
* <td align="center"> $Q_2$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0005.png
* </td> </tr> <tr> <td align="center"> $Q_2$ element, shape function 4 </td>
*
* <td align="center"> $Q_2$ element, shape function 5 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0006.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0007.png
* </td> </tr> <tr> <td align="center"> $Q_2$ element, shape function 6 </td>
*
* <td align="center"> $Q_2$ element, shape function 7 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q2/Q2_shape0008.png
* </td>
*
* <td align="center"> </td> </tr> <tr> <td align="center"> $Q_2$ element,
* shape function 8 </td>
*
* <td align="center"> </td> </tr> </table>
*
*
* <h4>Q3 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0--2--3--1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2--10-11-3
* | |
* 5 14 15 7
* | |
* 4 12 13 6
* | |
* 0--8--9--1
* @endverbatim
* </ul>
*
* In 2d, these shape functions look as follows (the black plane corresponds
* to zero; negative shape function values may not be visible): <table> <tr>
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0001.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 0 </td>
*
* <td align="center"> $Q_3$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0003.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 2 </td>
*
* <td align="center"> $Q_3$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0005.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 4 </td>
*
* <td align="center"> $Q_3$ element, shape function 5 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0006.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0007.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 6 </td>
*
* <td align="center"> $Q_3$ element, shape function 7 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0008.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0009.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 8 </td>
*
* <td align="center"> $Q_3$ element, shape function 9 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0010.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0011.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 10 </td>
*
* <td align="center"> $Q_3$ element, shape function 11 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0012.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0013.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 12 </td>
*
* <td align="center"> $Q_3$ element, shape function 13 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0014.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q3/Q3_shape0015.png
* </td> </tr> <tr> <td align="center"> $Q_3$ element, shape function 14 </td>
*
* <td align="center"> $Q_3$ element, shape function 15 </td> </tr> </table>
*
*
* <h4>Q4 elements</h4>
* <ul>
* <li> 1D case:
* @verbatim
* 0--2--3--4--1
* @endverbatim
*
* <li> 2D case:
* @verbatim
* 2--13-14-15-3
* | |
* 6 22 23 24 9
* | |
* 5 19 20 21 8
* | |
* 4 16 17 18 7
* | |
* 0--10-11-12-1
* @endverbatim
* </ul>
*
* In 2d, these shape functions look as follows (the black plane corresponds
* to zero; negative shape function values may not be visible): <table> <tr>
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0000.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0001.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 0 </td>
*
* <td align="center"> $Q_4$ element, shape function 1 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0002.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0003.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 2 </td>
*
* <td align="center"> $Q_4$ element, shape function 3 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0004.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0005.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 4 </td>
*
* <td align="center"> $Q_4$ element, shape function 5 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0006.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0007.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 6 </td>
*
* <td align="center"> $Q_4$ element, shape function 7 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0008.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0009.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 8 </td>
*
* <td align="center"> $Q_4$ element, shape function 9 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0010.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0011.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 10 </td>
*
* <td align="center"> $Q_4$ element, shape function 11 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0012.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0013.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 12 </td>
*
* <td align="center"> $Q_4$ element, shape function 13 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0014.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0015.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 14 </td>
*
* <td align="center"> $Q_4$ element, shape function 15 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0016.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0017.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 16 </td>
*
* <td align="center"> $Q_4$ element, shape function 17 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0018.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0019.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 18 </td>
*
* <td align="center"> $Q_4$ element, shape function 19 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0020.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0021.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 20 </td>
*
* <td align="center"> $Q_4$ element, shape function 21 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0022.png
* </td>
*
* <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0023.png
* </td> </tr> <tr> <td align="center"> $Q_4$ element, shape function 22 </td>
*
* <td align="center"> $Q_4$ element, shape function 23 </td> </tr>
*
* <tr> <td align="center">
* @image html http://www.dealii.org/images/shape-functions/lagrange/Q4/Q4_shape0024.png
* </td>
*
* <td align="center"> </td> </tr> <tr> <td align="center"> $Q_4$ element,
* shape function 24 </td>
*
* <td align="center"> </td> </tr> </table>
*
*
*
* @author Wolfgang Bangerth, 1998, 2003; Guido Kanschat, 2001; Ralf Hartmann,
* 2001, 2004, 2005; Oliver Kayser-Herold, 2004; Katharina Kormann, 2008;
* Martin Kronbichler, 2008
*/
template <int dim, int spacedim=dim>
class FE_Q : public FE_Q_Base<TensorProductPolynomials<dim>,dim,spacedim>
{
public:
/**
* Constructor for tensor product polynomials of degree @p p.
*/
FE_Q (const unsigned int p);
/**
* Constructor for tensor product polynomials with support points @p points
* based on a one-dimensional quadrature formula. The degree of the finite
* element is <tt>points.size()-1</tt>. Note that the first point has to be
* 0 and the last one 1. If
* <tt>FE_Q<dim>(QGaussLobatto<1>(fe_degree+1))</tt> is specified, so-called
* Gauss-Lobatto elements are obtained which can give a diagonal mass matrix
* if combined with Gauss-Lobatto quadrature on the same points. Their use
* is shown in step-48.
*/
FE_Q (const Quadrature<1> &points);
/**
* Constructs a FE_Q_isoQ1 element. That element shares large parts of code
* with FE_Q so most of the construction work is done in this routine,
* whereas the public constructor is in the class FE_Q_isoQ1.
*/
FE_Q(const unsigned int subdivisions_per_dimension,
const unsigned int base_degree);
/**
* Return a string that uniquely identifies a finite element. This class
* returns <tt>FE_Q<dim>(degree)</tt>, with @p dim and @p degree replaced by
* appropriate values.
*/
virtual std::string get_name () const;
protected:
/**
* @p clone function instead of a copy constructor.
*
* This function is needed by the constructors of @p FESystem.
*/
virtual FiniteElement<dim,spacedim> *clone() const;
};
/*@}*/
DEAL_II_NAMESPACE_CLOSE
#endif
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