poisson2d_4.h

Dolfin provide a high-performance linear algebra library

  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 < 1*num_derivatives; j++)
      values[j] = 0;
    
    // 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);
    const double scalings_y_2 = scalings_y_1*(0.5 - 0.5*y);
    const double scalings_y_3 = scalings_y_2*(0.5 - 0.5*y);
    const double scalings_y_4 = scalings_y_3*(0.5 - 0.5*y);
    
    // Compute psitilde_a
    const double psitilde_a_0 = 1;
    const double psitilde_a_1 = x;
    const double psitilde_a_2 = 1.5*x*psitilde_a_1 - 0.5*psitilde_a_0;
    const double psitilde_a_3 = 1.66666666666667*x*psitilde_a_2 - 0.666666666666667*psitilde_a_1;
    const double psitilde_a_4 = 1.75*x*psitilde_a_3 - 0.75*psitilde_a_2;
    
    // 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_0_2 = 0.111111111111111*psitilde_bs_0_1 + 1.66666666666667*y*psitilde_bs_0_1 - 0.555555555555556*psitilde_bs_0_0;
    const double psitilde_bs_0_3 = 0.05*psitilde_bs_0_2 + 1.75*y*psitilde_bs_0_2 - 0.7*psitilde_bs_0_1;
    const double psitilde_bs_0_4 = 0.0285714285714286*psitilde_bs_0_3 + 1.8*y*psitilde_bs_0_3 - 0.771428571428571*psitilde_bs_0_2;
    const double psitilde_bs_1_0 = 1;
    const double psitilde_bs_1_1 = 2.5*y + 1.5;
    const double psitilde_bs_1_2 = 0.54*psitilde_bs_1_1 + 2.1*y*psitilde_bs_1_1 - 0.56*psitilde_bs_1_0;
    const double psitilde_bs_1_3 = 0.285714285714286*psitilde_bs_1_2 + 2*y*psitilde_bs_1_2 - 0.714285714285714*psitilde_bs_1_1;
    const double psitilde_bs_2_0 = 1;
    const double psitilde_bs_2_1 = 3.5*y + 2.5;
    const double psitilde_bs_2_2 = 1.02040816326531*psitilde_bs_2_1 + 2.57142857142857*y*psitilde_bs_2_1 - 0.551020408163265*psitilde_bs_2_0;
    const double psitilde_bs_3_0 = 1;
    const double psitilde_bs_3_1 = 4.5*y + 3.5;
    const double psitilde_bs_4_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;
    const double basisvalue3 = 2.73861278752583*psitilde_a_2*scalings_y_2*psitilde_bs_2_0;
    const double basisvalue4 = 2.12132034355964*psitilde_a_1*scalings_y_1*psitilde_bs_1_1;
    const double basisvalue5 = 1.22474487139159*psitilde_a_0*scalings_y_0*psitilde_bs_0_2;
    const double basisvalue6 = 3.74165738677394*psitilde_a_3*scalings_y_3*psitilde_bs_3_0;
    const double basisvalue7 = 3.16227766016838*psitilde_a_2*scalings_y_2*psitilde_bs_2_1;
    const double basisvalue8 = 2.44948974278318*psitilde_a_1*scalings_y_1*psitilde_bs_1_2;
    const double basisvalue9 = 1.4142135623731*psitilde_a_0*scalings_y_0*psitilde_bs_0_3;
    const double basisvalue10 = 4.74341649025257*psitilde_a_4*scalings_y_4*psitilde_bs_4_0;
    const double basisvalue11 = 4.18330013267038*psitilde_a_3*scalings_y_3*psitilde_bs_3_1;
    const double basisvalue12 = 3.53553390593274*psitilde_a_2*scalings_y_2*psitilde_bs_2_2;
    const double basisvalue13 = 2.73861278752583*psitilde_a_1*scalings_y_1*psitilde_bs_1_3;
    const double basisvalue14 = 1.58113883008419*psitilde_a_0*scalings_y_0*psitilde_bs_0_4;
    
    // Table(s) of coefficients
    const static double coefficients0[15][15] = \
    {{0, -0.0412393049421161, -0.0238095238095238, 0.0289800294976278, 0.0224478343233825, 0.012960263189329, -0.0395942580610999, -0.0334632556631574, -0.025920526378658, -0.014965222882255, 0.0321247254366312, 0.0283313448138523, 0.023944356611608, 0.0185472188784818, 0.0107082418122104},
    {0, 0.0412393049421161, -0.0238095238095238, 0.0289800294976279, -0.0224478343233824, 0.012960263189329, 0.0395942580610999, -0.0334632556631574, 0.025920526378658, -0.014965222882255, 0.0321247254366312, -0.0283313448138523, 0.0239443566116079, -0.0185472188784818, 0.0107082418122104},
    {0, 0, 0.0476190476190476, 0, 0, 0.038880789567987, 0, 0, 0, 0.0598608915290199, 0, 0, 0, 0, 0.0535412090610519},
    {0.125707872210942, 0.131965775814772, -0.0253968253968254, 0.139104141588614, -0.0718330698348239, 0.0311046316543895, 0.0633508128977599, 0.0267706045305259, -0.0622092633087791, 0.0478887132232159, 0, 0.0566626896277046, -0.0838052481406278, 0.0834624849531682, -0.0535412090610519},
    {-0.0314269680527354, 0.0109971479845643, 0.00634920634920635, 0, 0.188561808316413, -0.163299316185545, 0, 0.0936971158568409, 0, -0.0419026240703139, 0, 0, 0.0838052481406278, -0.139104141588614, 0.107082418122104},
    {0.125707872210942, 0.0439885919382572, 0.126984126984127, 0, 0.035916534917412, 0.155523158271948, 0, 0, 0.103682105514632, -0.011972178305804, 0, 0, 0, 0.0927360943924091, -0.107082418122104},
    {0.125707872210942, -0.131965775814772, -0.0253968253968254, 0.139104141588614, 0.0718330698348239, 0.0311046316543895, -0.0633508128977599, 0.026770604530526, 0.0622092633087791, 0.0478887132232159, 0, -0.0566626896277046, -0.0838052481406278, -0.0834624849531682, -0.0535412090610519},
    {-0.0314269680527353, -0.0109971479845645, 0.00634920634920637, 0, -0.188561808316413, -0.163299316185545, 0, 0.0936971158568409, 0, -0.0419026240703139, 0, 0, 0.0838052481406278, 0.139104141588614, 0.107082418122104},
    {0.125707872210942, -0.0439885919382572, 0.126984126984127, 0, -0.0359165349174119, 0.155523158271948, 0, 0, -0.103682105514632, -0.011972178305804, 0, 0, 0, -0.0927360943924091, -0.107082418122104},
    {0.125707872210942, -0.0879771838765144, -0.101587301587302, 0.0927360943924091, 0.107749604752236, 0.0725774738602423, 0.0791885161221998, -0.013385302265263, -0.0518410527573159, -0.0419026240703139, -0.128498901746525, -0.0566626896277046, -0.011972178305804, 0.00927360943924091, 0.0107082418122104},
    {-0.0314269680527354, 0, -0.0126984126984127, -0.243432247780074, 0, 0.0544331053951818, 0, 0.0936971158568409, 0, -0.0419026240703139, 0.192748352619787, 0, -0.023944356611608, 0, 0.0107082418122104},
    {0.125707872210942, 0.0879771838765144, -0.101587301587302, 0.0927360943924091, -0.107749604752236, 0.0725774738602423, -0.0791885161221998, -0.013385302265263, 0.051841052757316, -0.0419026240703139, -0.128498901746525, 0.0566626896277046, -0.011972178305804, -0.0092736094392409, 0.0107082418122104},
    {0.251415744421884, -0.351908735506058, -0.203174603174603, -0.139104141588614, -0.107749604752236, -0.0622092633087791, 0.19005243869328, -0.0267706045305259, 0.124418526617558, 0.155638317975452, 0, 0.169988068883114, 0.0838052481406278, -0.0278208283177228, -0.0535412090610519},
    {0.251415744421884, 0.351908735506058, -0.203174603174603, -0.139104141588614, 0.107749604752236, -0.0622092633087791, -0.19005243869328, -0.0267706045305259, -0.124418526617558, 0.155638317975452, 0, -0.169988068883114, 0.0838052481406278, 0.0278208283177227, -0.0535412090610519},
    {0.251415744421883, 0, 0.406349206349206, 0, 0, -0.186627789926337, 0, -0.187394231713682, 0, -0.203527031198668, 0, 0, -0.167610496281256, 0, 0.107082418122104}};
    
    // Interesting (new) part
    // Tables of derivatives of the polynomial base (transpose)
    const static double dmats0[15][15] = \
    {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {2.44948974278318, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 4.74341649025257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {2, 0, 3.53553390593274, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {2.64575131106459, 0, -1.49666295470958, 6.83130051063973, 0, 0.305505046330389, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 2.19089023002066, 0, 0, 6.26099033699941, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1.73205080756888, 0, 3.91918358845309, 0, 0, 4.20000000000001, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 5.47722557505166, 0, 0, -1.91662969499982, 0, 8.87411967464942, 0, 0.276641667586244, 0, 0, 0, 0, 0, 0},
    {2.36643191323984, 0, 1.67332005306815, 2.18217890235992, 0, -2.53734018966619, 0, 8.50420064270761, 0, 0.760638829255663, 0, 0, 0, 0, 0},
    {0, 1.22474487139159, 0, 0, 4.57142857142857, 0, 0, 0, 7.4230748895809, 0, 0, 0, 0, 0, 0},
    {1.54919333848297, 0, 3.83405790253616, 0, 0, 5.3665631459995, 0, 0, 0, 4.6475800154489, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}};
    
    const static double dmats1[15][15] = \
    {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1.22474487