polynomialspacedimension.hpp

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// Core data structures and methods required to implement the discrete de Rham sequence in 3D
//
// Provides:
//  - Dimension of ull and partial polynomial spaces on the element, faces, and edges
//
// Author: Daniele Di Pietro (daniele.di-pietro@umontpellier.fr)
//
 
/*
 *
 * This library was developed around HHO methods, although some parts of it have a more
 * general purpose. If you use this code or part of it in a scientific publication, 
 * please mention the following book as a reference for the underlying principles
 * of HHO schemes:
 *
 * The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications. 
 *  D. A. Di Pietro and J. Droniou. Modeling, Simulation and Applications, vol. 19. 
 *  Springer International Publishing, 2020, xxxi + 525p. doi: 10.1007/978-3-030-37203-3. 
 *  url: https://hal.archives-ouvertes.fr/hal-02151813.
 *
 */
 
/*
 * The DDR sequence has been designed in
 *
 *  Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra.
 *   D. A. Di Pietro, J. Droniou, and F. Rapetti, 33p, 2019. url: https://arxiv.org/abs/1911.03616.
 *
 * If you use this code in a scientific publication, please mention the above article.
 *
 */
#ifndef POLYNOMIALSPACEDIMENSION_HPP
#define POLYNOMIALSPACEDIMENSION_HPP
 
#include <mesh.hpp>
 
namespace HArDCore3D
{
 
  template<typename GeometricSupport>
  struct PolynomialSpaceDimension
  {
    // Only specializations are relevant
  };
  
  template<>
  struct PolynomialSpaceDimension<Cell>
  {
    static size_t Poly(int k)
    {
      return (k >= 0 ? (k+1)*(k+2)*(k+3)/6 : 0);
    }
    static size_t Goly(int k)
    {
      return (k >= 0 ? PolynomialSpaceDimension<Cell>::Poly(k + 1) - 1 : 0);
    }
    static size_t GolyCompl(int k)
    {
      return 3 * PolynomialSpaceDimension<Cell>::Poly(k) - PolynomialSpaceDimension<Cell>::Goly(k);
    }
    static size_t Roly(int k)
    {
      return (k >= 0 ? PolynomialSpaceDimension<Cell>::GolyCompl(k + 1) : 0);
    }
    static size_t RolyCompl(int k)
    {
      return 3 * PolynomialSpaceDimension<Cell>::Poly(k) - PolynomialSpaceDimension<Cell>::Roly(k);
    }
  };
 
  template<>
  struct PolynomialSpaceDimension<Face>
  {
    static size_t Poly(int k)
    {
      return (k >= 0 ? (k + 1) * (k + 2) / 2 : 0);
    }
    static size_t Goly(int k)
    {
      return (k >= 0 ? PolynomialSpaceDimension<Face>::Poly(k + 1) - 1 : 0);
    }
    static size_t GolyCompl(int k)
    {
      return 2 * PolynomialSpaceDimension<Face>::Poly(k) - PolynomialSpaceDimension<Face>::Goly(k);
    }
    static size_t Roly(int k)
    {
      return (k >= 0 ? PolynomialSpaceDimension<Face>::Poly(k + 1) - 1 : 0);
    }
    static size_t RolyCompl(int k)
    {
      return 2 * PolynomialSpaceDimension<Face>::Poly(k) - PolynomialSpaceDimension<Face>::Roly(k);
    }
  };
 
  template<>
  struct PolynomialSpaceDimension<Edge>
  {
    static size_t Poly(int k)
    {
      return (k >= 0 ? k + 1 : 0);
    }
  };
 
  
} // namespace HArDCore3D
#endif