mixedpoisson.h

Dolfin provide a high-performance linear algebra library

        }
    
      }
      // Compute derivatives on reference element as dot product of coefficients and basisvalues
      derivatives[deriv_num] = new_coeff0_0*basisvalue0;
    }
    
    // Transform derivatives back to physical element
    for (unsigned int row = 0; row < num_derivatives; row++)
    {
      for (unsigned int col = 0; col < num_derivatives; col++)
      {
        values[row] += transform[row][col]*derivatives[col];
      }
    }
    // Delete pointer to array of derivatives on FIAT element
    delete [] derivatives;
    
    // Delete pointer to array of combinations of derivatives
    delete [] combinations;
    
  }

  /// Evaluate linear functional for dof i on the function f
  virtual double evaluate_dof(unsigned int i,
                              const ufc::function& f,
                              const ufc::cell& c) const
  {
    double values[1];
    double coordinates[2];
    
    // Nodal coordinates on reference cell
    static double X[1][2] = {{0.333333333333333, 0.333333333333333}};
    
    // Components for each dof
    static unsigned int components[1] = {0};
    
    // Extract vertex coordinates
    const double * const * x = c.coordinates;
    
    // Evaluate basis functions for affine mapping
    const double w0 = 1.0 - X[i][0] - X[i][1];
    const double w1 = X[i][0];
    const double w2 = X[i][1];
    
    // Compute affine mapping x = F(X)
    coordinates[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
    coordinates[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
    
    // Evaluate function at coordinates
    f.evaluate(values, coordinates, c);
    
    // Pick component for evaluation
    return values[components[i]];
  }

  /// Interpolate vertex values from dof values
  virtual void interpolate_vertex_values(double* vertex_values,
                                         const double* dof_values,
                                         const ufc::cell& c) const
  {
    // Evaluate at vertices and use affine mapping
    vertex_values[0] = dof_values[0];
    vertex_values[1] = dof_values[0];
    vertex_values[2] = dof_values[0];
  }

  /// Return the number of sub elements (for a mixed element)
  virtual unsigned int num_sub_elements() const
  {
    return 1;
  }

  /// Create a new finite element for sub element i (for a mixed element)
  virtual ufc::finite_element* create_sub_element(unsigned int i) const
  {
    return new UFC_MixedPoissonBilinearForm_finite_element_0_1();
  }

};

/// This class defines the interface for a finite element.

class UFC_MixedPoissonBilinearForm_finite_element_0: public ufc::finite_element
{
public:

  /// Constructor
  UFC_MixedPoissonBilinearForm_finite_element_0() : ufc::finite_element()
  {
    // Do nothing
  }

  /// Destructor
  virtual ~UFC_MixedPoissonBilinearForm_finite_element_0()
  {
    // Do nothing
  }

  /// Return a string identifying the finite element
  virtual const char* signature() const
  {
    return "Mixed finite element: [Brezzi-Douglas-Marini finite element of degree 1 on a triangle, Discontinuous Lagrange finite element of degree 0 on a triangle]";
  }

  /// Return the cell shape
  virtual ufc::shape cell_shape() const
  {
    return ufc::triangle;
  }

  /// Return the dimension of the finite element function space
  virtual unsigned int space_dimension() const
  {
    return 7;
  }

  /// Return the rank of the value space
  virtual unsigned int value_rank() const
  {
    return 1;
  }

  /// Return the dimension of the value space for axis i
  virtual unsigned int value_dimension(unsigned int i) const
  {
    return 3;
  }

  /// Evaluate basis function i at given point in cell
  virtual void evaluate_basis(unsigned int i,
                              double* values,
                              const double* coordinates,
                              const ufc::cell& c) const
  {
    // Extract vertex coordinates
    const double * const * element_coordinates = c.coordinates;
    
    // Compute Jacobian of affine map from reference cell
    const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
    const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
    const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
    const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
      
    // Compute determinant of Jacobian
    const double detJ = J_00*J_11 - J_01*J_10;
    
    // Compute constants
    const double C0 = element_coordinates[1][0] + element_coordinates[2][0];
    const double C1 = element_coordinates[1][1] + element_coordinates[2][1];
    
    // Get coordinates and map to the reference (FIAT) element
    double x = (J_01*C1 - J_11*C0 + 2.0*J_11*coordinates[0] - 2.0*J_01*coordinates[1]) / detJ;
    double y = (J_10*C0 - J_00*C1 - 2.0*J_10*coordinates[0] + 2.0*J_00*coordinates[1]) / detJ;
    
    // Map coordinates to the reference square
    if (std::abs(y - 1.0) < 1e-14)
      x = -1.0;
    else
      x = 2.0 * (1.0 + x)/(1.0 - y) - 1.0;
    
    // Reset values
    values[0] = 0;
    values[1] = 0;
    values[2] = 0;
    
    if (0 <= i and i <= 5)
    {
      // Map degree of freedom to element degree of freedom
      const unsigned int dof = i;
    
      // Generate scalings
      const double scalings_y_0 = 1;
      const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
      const double psitilde_a_1 = x;
    
      // Compute psitilde_bs
      const double psitilde_bs_0_0 = 1;
      const double psitilde_bs_0_1 = 1.5*y + 0.5;
      const double psitilde_bs_1_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
      const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
      const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
    
      // Table(s) of coefficients
      const static double coefficients0[6][3] =   \
      {{0.942809041582063, 0.577350269189626, -0.333333333333333},
      {-0.471404520791032, -0.288675134594813, 0.166666666666667},
      {0.471404520791032, -0.577350269189626, -0.666666666666667},
      {0.471404520791032, 0.288675134594813, 0.833333333333333},
      {-0.471404520791032, -0.288675134594813, 0.166666666666667},
      {0.942809041582063, 0.577350269189626, -0.333333333333333}};
    
      const static double coefficients1[6][3] =   \
      {{-0.471404520791032, 0, -0.333333333333333},
      {0.942809041582063, 0, 0.666666666666667},
      {0.471404520791032, 0, 0.333333333333333},
      {-0.942809041582063, 0, -0.666666666666667},
      {-0.471404520791032, 0.866025403784439, 0.166666666666667},
      {-0.471404520791032, -0.866025403784439, 0.166666666666667}};
    
      // Extract relevant coefficients
      const double coeff0_0 =   coefficients0[dof][0];
      const double coeff0_1 =   coefficients0[dof][1];
      const double coeff0_2 =   coefficients0[dof][2];
      const double coeff1_0 =   coefficients1[dof][0];
      const double coeff1_1 =   coefficients1[dof][1];
      const double coeff1_2 =   coefficients1[dof][2];
    
      // Compute value(s)
      values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
      values[1] = coeff1_0*basisvalue0 + coeff1_1*basisvalue1 + coeff1_2*basisvalue2;
    }
    
    if (6 <= i and i <= 6)
    {
      // Map degree of freedom to element degree of freedom
      const unsigned int dof = i - 6;
    
      // Generate scalings
      const double scalings_y_0 = 1;
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
    
      // Compute psitilde_bs
      const double psitilde_bs_0_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
    
      // Table(s) of coefficients
      const static double coefficients0[1][1] =   \
      {{1.41421356237309}};
    
      // Extract relevant coefficients
      const double coeff0_0 =   coefficients0[dof][0];
    
      // Compute value(s)
      values[2] = coeff0_0*basisvalue0;
    }
    
  }

  /// Evaluate order n derivatives of basis function i at given point in cell
  virtual void evaluate_basis_derivatives(unsigned int i,
                                          unsigned int n,
                                          double* values,
                                          const double* coordinates,
                                          const ufc::cell& c) const
  {
    // Extract vertex coordinates
    const double * const * element_coordinates = c.coordinates;
    
    // Compute Jacobian of affine map from reference cell
    const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
    const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
    const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
    const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
      
    // Compute determinant of Jacobian
    const double detJ = J_00*J_11 - J_01*J_10;
    
    // Compute constants
    const double C0 = element_coordinates[1][0] + element_coordinates[2][0];
    const double C1 = element_coordinates[1][1] + element_coordinates[2][1];
    
    // Get coordinates and map to the reference (FIAT) element
    double x = (J_01*C1 - J_11*C0 + 2.0*J_11*coordinates[0] - 2.0*J_01*coordinates[1]) / detJ;
    double y = (J_10*C0 - J_00*C1 - 2.0*J_10*coordinates[0] + 2.0*J_00*coordinates[1]) / detJ;
    
    // Map coordinates to the reference square
    if (std::abs(y - 1.0) < 1e-14)
      x = -1.0;
    else
      x = 2.0 * (1.0 + x)/(1.0 - y) - 1.0;
    
    // Compute number of derivatives
    unsigned int num_derivatives = 1;
    
    for (unsigned int j = 0; j < n; j++)
      num_derivatives *= 2;
    
    
    // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
    unsigned int **combinations = new unsigned int *[num_derivatives];
        
    for (unsigned int j = 0; j < num_derivatives; j++)
    {
      combinations[j] = new unsigned int [n];
      for (unsigned int k = 0; k < n; k++)
        combinations[j][k] = 0;
    }
        
    // Generate combinations of derivatives
    for (unsigned int row = 1; row < num_derivatives; row++)
    {
      for (unsigned int num = 0; num < row; num++)
      {
        for (unsigned int col = n-1; col+1 > 0; col--)
        {
          if (combinations[row][col] + 1 > 1)
            combinations[row][col] = 0;
          else
          {
            combinations[row][col] += 1;
            break;
          }
        }
      }
    }
    
    // Compute inverse of Jacobian, components are scaled because of difference in FFC/FIAT reference elements
    const double Jinv[2][2] =  {{2*J_11 / detJ, -2*J_01 / detJ}, {-2*J_10 / detJ, 2*J_00 / detJ}};
    
    // Declare transformation matrix
    // Declare pointer to two dimensional array and initialise
    double **transform = new double *[num_derivatives];
        
    for (unsigned int j = 0; j < num_derivatives; j++)
    {
      transform[j] = new double [num_derivatives];
      for (unsigned int k = 0; k < num_derivatives; k++)
        transform[j][k] = 1;
    }
    
    // Construct transformation matrix
    for (unsigned int row = 0; row < num_derivatives; row++)
    {
      for (unsigned int col = 0; col < num_derivatives; col++)
      {
        for (unsigned int k = 0; k < n; k++)
          transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
      }
    }
    
    // Reset values
    for (unsigned int j = 0; j < 3*num_derivatives; j++)
      values[j] = 0;
    
    if (0 <= i and i <= 5)
    {
      // Map degree of freedom to element degree of freedom
      const unsigned int dof = i;
    
      // Generate scalings
      const double scalings_y_0 = 1;
      const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
      const double psitilde_a_1 = x;