--- /dev/null
+/* $Id$ */
+
+#include <fe/fe_lib.h>
+#include <grid/tria_iterator.h>
+#include <grid/dof_accessor.h>
+#include <grid/geometry_info.h>
+#include <algorithm>
+
+
+/*--------------------------------- For 1d ---------------------------------
+ -- Use the following maple script to generate the basis functions,
+ -- gradients and prolongation matrices as well as the mass matrix.
+ -- Make sure that the files do not exists beforehand, since output
+ -- is appended instead of overwriting previous contents.
+ --
+ -- You should only have to change the very first lines for polynomials
+ -- of higher order.
+ --------------------------------------------------------------------------
+ n_functions := 4;
+
+ ansatz_points := array(0..n_functions-1);
+ ansatz_points[0] := 0;
+ ansatz_points[1] := 1;
+ ansatz_points[2] := 1/3;
+ ansatz_points[3] := 2/3;
+
+ phi_polynom := array(0..n_functions-1);
+ grad_phi_polynom := array(0..n_functions-1);
+ local_mass_matrix := array(0..n_functions-1, 0..n_functions-1);
+
+ for i from 0 to n_functions-1 do
+ # note that the interp function wants vector indexed from
+ # one and not from zero.
+ values := array(1..n_functions);
+ for j from 1 to n_functions do
+ values[j] := 0;
+ od;
+ values[i+1] := 1;
+
+ shifted_ansatz_points := array (1..n_functions);
+ for j from 1 to n_functions do
+ shifted_ansatz_points[j] := ansatz_points[j-1];
+ od;
+
+ phi_polynom[i] := interp (shifted_ansatz_points, values, xi);
+ grad_phi_polynom[i] := diff(phi_polynom[i], xi);
+ od;
+
+ phi:= proc(i,x) subs(xi=x, phi_polynom[i]); end;
+
+
+ points[0] := array(0..n_functions-1);
+ points[1] := array(0..n_functions-1);
+ for i from 0 to n_functions-1 do
+ points[0][i] := ansatz_points[i]/2;
+ points[1][i] := ansatz_points[i]/2+1/2;
+ od;
+
+ prolongation := array(0..1,0..n_functions-1, 0..n_functions-1);
+
+ for i from 0 to 1 do
+ for j from 0 to n_functions-1 do
+ for k from 0 to n_functions-1 do
+ prolongation[i,j,k] := phi(k, points[i][j]);
+ od;
+ od;
+ od;
+
+ for i from 0 to n_functions-1 do
+ for j from 0 to n_functions-1 do
+ local_mass_matrix[i,j] := int(phi_polynom[i] * phi_polynom[j] * h,
+ xi=0..1);
+ od;
+ od;
+
+ readlib(C);
+ C(phi_polynom, filename=shape_value_1d);
+ C(grad_phi_polynom, filename=shape_grad_1d);
+ C(prolongation, filename=prolongation_1d);
+ C(local_mass_matrix, optimized, filename=massmatrix_1d);
+
+ -----------------------------------------------------------------------
+ Use the following perl scripts to convert the output into a
+ suitable format:
+
+ perl -pi -e 's/phi_polynom\[(\d)\] =/case $1: return/g;' shape_value_1d
+ perl -pi -e 's/grad_phi_polynom\[(\d)\] = (.*);/case $1: return Point<1>($2);/g;' shape_grad_1d
+ perl -pi -e 's/\[(\d)\]\[(\d)\]/($1,$2)/g;' massmatrix_1d
+ perl -pi -e 's/\[(\d)\]\[(\d)\]\[(\d)\]/[$1]($2,$3)/g;' prolongation_1d
+ perl -pi~ -e 's/(t\d)/const double $1/g;' massmatrix_1d
+*/
+
+
+
+
+/*--------------------------------- For 2d ---------------------------------
+ -- Use the following maple script to generate the basis functions,
+ -- gradients and prolongation matrices as well as the mass matrix.
+ -- Make sure that the files do not exists beforehand, since output
+ -- is appended instead of overwriting previous contents.
+ --
+ -- You should only have to change the very first lines for polynomials
+ -- of higher order.
+ --------------------------------------------------------------------------
+ n_functions := 16;
+
+ ansatz_function := (a1 + a2*xi + a3*xi*xi + a4*xi*xi*xi) +
+ (b1 + b2*xi + b3*xi*xi + b4*xi*xi*xi)*eta +
+ (c1 + c2*xi + c3*xi*xi + c4*xi*xi*xi)*eta*eta +
+ (d1 + d2*xi + d3*xi*xi + d4*xi*xi*xi)*eta*eta*eta;
+ ansatz_points := array(0..n_functions-1);
+ # note: ansatz_points[i] is a vector which is indexed from
+ # one and not from zero!
+ ansatz_points[0] := [0,0];
+ ansatz_points[1] := [1,0];
+ ansatz_points[2] := [1,1];
+ ansatz_points[3] := [0,1];
+ ansatz_points[4] := [1/3,0];
+ ansatz_points[5] := [2/3,0];
+ ansatz_points[6] := [1,1/3];
+ ansatz_points[7] := [1,2/3];
+ ansatz_points[8] := [1/3,1];
+ ansatz_points[9] := [2/3,1];
+ ansatz_points[10]:= [0,1/3];
+ ansatz_points[11]:= [0,2/3];
+ ansatz_points[12]:= [1/3,1/3];
+ ansatz_points[13]:= [2/3,1/3];
+ ansatz_points[14]:= [2/3,2/3];
+ ansatz_points[15]:= [1/3,2/3];
+
+
+ phi_polynom := array(0..n_functions-1);
+ grad_phi_polynom := array(0..n_functions-1,0..1);
+ local_mass_matrix := array(0..n_functions-1, 0..n_functions-1);
+ prolongation := array(0..3,0..n_functions-1, 0..n_functions-1);
+
+
+ for i from 0 to n_functions-1 do
+ values := array(1..n_functions);
+ for j from 1 to n_functions do
+ values[j] := 0;
+ od;
+ values[i+1] := 1;
+
+ equation_system := {};
+ for j from 0 to n_functions-1 do
+ poly := subs(xi=ansatz_points[j][1],
+ eta=ansatz_points[j][2],
+ ansatz_function);
+ if (i=j) then
+ equation_system := equation_system union {poly = 1};
+ else
+ equation_system := equation_system union {poly = 0};
+ fi;
+ od;
+
+ phi_polynom[i] := subs(solve(equation_system), ansatz_function);
+ grad_phi_polynom[i,0] := diff(phi_polynom[i], xi);
+ grad_phi_polynom[i,1] := diff(phi_polynom[i], eta);
+ od;
+
+ phi:= proc(i,x,y) subs(xi=x, eta=y, phi_polynom[i]); end;
+
+ #points on children; let them be indexed one-based, as are
+ #the ansatz_points
+ points[0] := array(0..n_functions-1, 1..2);
+ points[1] := array(0..n_functions-1, 1..2);
+ points[2] := array(0..n_functions-1, 1..2);
+ points[3] := array(0..n_functions-1, 1..2);
+ for i from 0 to n_functions-1 do
+ points[0][i,1] := ansatz_points[i][1]/2;
+ points[0][i,2] := ansatz_points[i][2]/2;
+
+ points[1][i,1] := ansatz_points[i][1]/2+1/2;
+ points[1][i,2] := ansatz_points[i][2]/2;
+
+ points[2][i,1] := ansatz_points[i][1]/2+1/2;
+ points[2][i,2] := ansatz_points[i][2]/2+1/2;
+
+ points[3][i,1] := ansatz_points[i][1]/2;
+ points[3][i,2] := ansatz_points[i][2]/2+1/2;
+ od;
+
+ for i from 0 to 3 do
+ for j from 0 to n_functions-1 do
+ for k from 0 to n_functions-1 do
+ prolongation[i,j,k] := phi(k, points[i][j,1], points[i][j,2]);
+ od;
+ od;
+ od;
+
+ # tphi are the basis functions of the linear element. These functions
+ # are used for the computation of the subparametric transformation from
+ # unit cell to real cell.
+ tphi[0] := (1-xi)*(1-eta);
+ tphi[1] := xi*(1-eta);
+ tphi[2] := xi*eta;
+ tphi[3] := (1-xi)*eta;
+ x_real := sum(x[s]*tphi[s], s=0..3);
+ y_real := sum(y[s]*tphi[s], s=0..3);
+ detJ := diff(x_real,xi)*diff(y_real,eta) - diff(x_real,eta)*diff(y_real,xi);
+ for i from 0 to n_functions-1 do
+ for j from 0 to n_functions-1 do
+ local_mass_matrix[i,j] := int(int(phi_polynom[i] * phi_polynom[j] * detJ,
+ xi=0..1), eta=0..1);
+ od;
+ od;
+
+ readlib(C);
+ C(phi_polynom, filename=shape_value_2d);
+ C(grad_phi_polynom, filename=shape_grad_2d);
+ C(prolongation, filename=prolongation_2d);
+ C(local_mass_matrix, optimized, filename=massmatrix_2d);
+
+ -----------------------------------------------------------------------
+ Use the following perl scripts to convert the output into a
+ suitable format.
+
+ perl -pi -e 's/phi_polynom\[(\d+)\] =/case $1: return/g;' shape_value_2d
+ perl -pi -e 's/([^;])\n/$1/g;' shape_grad_2d
+ perl -pi -e 's/grad_phi_polynom\[(\d+)\]\[0\] = (.*);/case $1: return Point<2>($2,/g;' shape_grad_2d
+ perl -pi -e 's/grad_phi_polynom\[(\d+)\]\[1\] = (.*);/$2);/g;' shape_grad_2d
+ perl -pi -e 's/\[(\d+)\]\[(\d+)\]/($1,$2)/g;' massmatrix_2d
+ perl -pi -e 's/\[(\d+)\]\[(\d+)\]\[(\d+)\]/[$1]($2,$3)/g;' prolongation_2d
+ perl -pi~ -e 's/(t\d+) =/const double $1 =/g;' massmatrix_2d
+*/
+
+
+
+
+
+
+#if deal_II_dimension == 1
+
+template <>
+FECubicSub<1>::FECubicSub () :
+ FiniteElement<1> (1, 2) {
+ prolongation[0](0,0) = 1.0;
+ prolongation[0](0,1) = 0.0;
+ prolongation[0](0,2) = 0.0;
+ prolongation[0](0,3) = 0.0;
+ prolongation[0](1,0) = -1.0/16.0;
+ prolongation[0](1,1) = -1.0/16.0;
+ prolongation[0](1,2) = 9.0/16.0;
+ prolongation[0](1,3) = 9.0/16.0;
+ prolongation[0](2,0) = 5.0/16.0;
+ prolongation[0](2,1) = 1.0/16.0;
+ prolongation[0](2,2) = 15.0/16.0;
+ prolongation[0](2,3) = -5.0/16.0;
+ prolongation[0](3,0) = 0.0;
+ prolongation[0](3,1) = 0.0;
+ prolongation[0](3,2) = 1.0;
+ prolongation[0](3,3) = 0.0;
+ prolongation[1](0,0) = -1.0/16.0;
+ prolongation[1](0,1) = -1.0/16.0;
+ prolongation[1](0,2) = 9.0/16.0;
+ prolongation[1](0,3) = 9.0/16.0;
+ prolongation[1](1,0) = 0.0;
+ prolongation[1](1,1) = 1.0;
+ prolongation[1](1,2) = 0.0;
+ prolongation[1](1,3) = 0.0;
+ prolongation[1](2,0) = 0.0;
+ prolongation[1](2,1) = 0.0;
+ prolongation[1](2,2) = 0.0;
+ prolongation[1](2,3) = 1.0;
+ prolongation[1](3,0) = 1.0/16.0;
+ prolongation[1](3,1) = 5.0/16.0;
+ prolongation[1](3,2) = -5.0/16.0;
+ prolongation[1](3,3) = 15.0/16.0;
+};
+
+
+
+template <>
+void FECubicSub<1>::fill_fe_values (const DoFHandler<1>::cell_iterator &cell,
+ const vector<Point<1> > &unit_points,
+ vector<dFMatrix> &jacobians,
+ const bool compute_jacobians,
+ vector<Point<1> > &ansatz_points,
+ const bool compute_ansatz_points,
+ vector<Point<1> > &q_points,
+ const bool compute_q_points,
+ const Boundary<1> &boundary) const {
+ // simply pass down
+ FiniteElement<1>::fill_fe_values (cell, unit_points,
+ jacobians, compute_jacobians,
+ ansatz_points, compute_ansatz_points,
+ q_points, compute_q_points, boundary);
+};
+
+
+
+template <>
+double
+FECubicSub<1>::shape_value(const unsigned int i,
+ const Point<1> &p) const
+{
+ Assert((i<total_dofs), ExcInvalidIndex(i));
+ const double xi = p(0);
+ switch (i)
+ {
+ case 0: return -9.0/2.0*xi*xi*xi+9.0*xi*xi-11.0/2.0*xi+1.0;
+ case 1: return 9.0/2.0*xi*xi*xi-9.0/2.0*xi*xi+xi;
+ case 2: return 27.0/2.0*xi*xi*xi-45.0/2.0*xi*xi+9.0*xi;
+ case 3: return -27.0/2.0*xi*xi*xi+18.0*xi*xi-9.0/2.0*xi;
+ }
+ return 0.;
+};
+
+
+
+template <>
+inline
+double
+FECubicSub<1>::linear_shape_value(const unsigned int i,
+ const Point<1> &p) const
+{
+ Assert((i<2), ExcInvalidIndex(i));
+ const double xi = p(0);
+ switch (i)
+ {
+ case 0: return 1.-xi;
+ case 1: return xi;
+ }
+ return 0.;
+};
+
+
+
+template <>
+Point<1>
+FECubicSub<1>::shape_grad(const unsigned int i,
+ const Point<1> &p) const
+{
+ Assert((i<total_dofs), ExcInvalidIndex(i));
+ const double xi = p(0);
+ switch (i)
+ {
+ case 0: return Point<1>(-27.0/2.0*xi*xi+18.0*xi-11.0/2.0);
+ case 1: return Point<1>(27.0/2.0*xi*xi-9.0*xi+1.0);
+ case 2: return Point<1>(81.0/2.0*xi*xi-45.0*xi+9.0);
+ case 3: return Point<1>(-81.0/2.0*xi*xi+36.0*xi-9.0/2.0);
+ }
+ return Point<1>();
+};
+
+
+
+template <>
+inline
+Point<1>
+FECubicSub<1>::linear_shape_grad(const unsigned int i,
+ const Point<1>&) const
+{
+ Assert((i<2), ExcInvalidIndex(i));
+ switch (i)
+ {
+ case 0: return Point<1>(-1.);
+ case 1: return Point<1>(1.);
+ }
+ return Point<1>();
+};
+
+
+
+template <>
+void FECubicSub<1>::get_ansatz_points (const typename DoFHandler<1>::cell_iterator &cell,
+ const Boundary<1> &boundary,
+ vector<Point<1> > &ansatz_points) const {
+ FiniteElement<1>::get_ansatz_points (cell, boundary, ansatz_points);
+};
+
+
+
+template <>
+void FECubicSub<1>::get_face_ansatz_points (const typename DoFHandler<1>::face_iterator &,
+ const Boundary<1> &,
+ vector<Point<1> > &) const {
+ Assert (false, ExcInternalError());
+};
+
+
+
+template <>
+void FECubicSub<1>::get_face_jacobians (const DoFHandler<1>::face_iterator &,
+ const Boundary<1> &,
+ const vector<Point<0> > &,
+ vector<double> &) const {
+ Assert (false, ExcInternalError());
+};
+
+
+
+template <>
+void FECubicSub<1>::get_subface_jacobians (const DoFHandler<1>::face_iterator &,
+ const unsigned int ,
+ const vector<Point<0> > &,
+ vector<double> &) const {
+ Assert (false, ExcInternalError());
+};
+
+
+
+template <>
+void FECubicSub<1>::get_normal_vectors (const DoFHandler<1>::cell_iterator &,
+ const unsigned int,
+ const Boundary<1> &,
+ const vector<Point<0> > &,
+ vector<Point<1> > &) const {
+ Assert (false, ExcInternalError());
+};
+
+
+
+template <>
+void FECubicSub<1>::get_normal_vectors (const DoFHandler<1>::cell_iterator &,
+ const unsigned int,
+ const unsigned int,
+ const vector<Point<0> > &,
+ vector<Point<1> > &) const {
+ Assert (false, ExcInternalError());
+};
+
+
+
+template <>
+void FECubicSub<1>::get_local_mass_matrix (const DoFHandler<1>::cell_iterator &cell,
+ const Boundary<1> &,
+ dFMatrix &local_mass_matrix) const {
+ Assert (local_mass_matrix.n() == total_dofs,
+ ExcWrongFieldDimension(local_mass_matrix.n(),total_dofs));
+ Assert (local_mass_matrix.m() == total_dofs,
+ ExcWrongFieldDimension(local_mass_matrix.m(),total_dofs));
+
+ const double h = cell->vertex(1)(0) - cell->vertex(0)(0);
+ Assert (h>0, ExcJacobiDeterminantHasWrongSign());
+
+ const double t1 = 8.0/105.0*h;
+ const double t2 = 19.0/1680.0*h;
+ const double t3 = 33.0/560.0*h;
+ const double t4 = 3.0/140.0*h;
+ const double t5 = 27.0/70.0*h;
+ const double t6 = 27.0/560.0*h;
+ local_mass_matrix(0,0) = const double t1;
+ local_mass_matrix(0,1) = const double t2;
+ local_mass_matrix(0,2) = const double t3;
+ local_mass_matrix(0,3) = - const double t4;
+ local_mass_matrix(1,0) = const double t2;
+ local_mass_matrix(1,1) = const double t1;
+ local_mass_matrix(1,2) = - const double t4;
+ local_mass_matrix(1,3) = const double t3;
+ local_mass_matrix(2,0) = const double t3;
+ local_mass_matrix(2,1) = - const double t4;
+ local_mass_matrix(2,2) = const double t5;
+ local_mass_matrix(2,3) = - const double t6;
+ local_mass_matrix(3,0) = - const double t4;
+ local_mass_matrix(3,1) = const double t3;
+ local_mass_matrix(3,2) = - const double t6;
+ local_mass_matrix(3,3) = const double t5;
+};
+
+#endif
+
+
+
+
+#if deal_II_dimension == 2
+
+template <>
+FECubicSub<2>::FECubicSub () :
+ FiniteElement<2> (1, 1, 1)
+{
+ interface_constraints(0,2) = 1.0;
+ interface_constraints(1,0) = 3./8.;
+ interface_constraints(1,1) = -1./8.;
+ interface_constraints(1,2) = 3./4.;
+ interface_constraints(2,0) = -1./8.;
+ interface_constraints(2,1) = 3./8.;
+ interface_constraints(2,2) = 3./4.;
+
+/*
+ Get the prolongation matrices by the following little maple script:
+
+ phi[0] := proc(xi,eta) (1-xi)*( 2*xi-1) * (1-eta)*( 2*eta-1); end;
+ phi[1] := proc(xi,eta) xi *(-2*xi+1) * (1-eta)*( 2*eta-1); end;
+ phi[2] := proc(xi,eta) xi *(-2*xi+1) * eta *(-2*eta+1); end;
+ phi[3] := proc(xi,eta) (1-xi)*( 2*xi-1) * eta *(-2*eta+1); end;
+ phi[4] := proc(xi,eta) 4 * (1-xi)*xi * (1-eta)*(1-2*eta); end;
+ phi[5] := proc(xi,eta) 4 * xi *(-1+2*xi) * (1-eta)*eta; end;
+ phi[6] := proc(xi,eta) 4 * (1-xi)*xi * eta *(-1+2*eta);end;
+ phi[7] := proc(xi,eta) 4 * (1-xi)*(1-2*xi) * (1-eta)*eta; end;
+ phi[8] := proc(xi,eta) 16 * xi*(1-xi) * eta*(1-eta); end;
+
+ points_x[0] := array(0..8, [0, 1/2, 1/2, 0, 1/4, 1/2, 1/4, 0, 1/4]);
+ points_y[0] := array(0..8, [0, 0, 1/2, 1/2, 0, 1/4, 1/2, 1/4, 1/4]);
+
+ points_x[1] := array(0..8, [1/2, 1, 1, 1/2, 3/4, 1, 3/4, 1/2, 3/4]);
+ points_y[1] := array(0..8, [0, 0, 1/2, 1/2, 0, 1/4, 1/2, 1/4, 1/4]);
+
+ points_x[2] := array(0..8, [1/2, 1, 1, 1/2, 3/4, 1, 3/4, 1/2, 3/4]);
+ points_y[2] := array(0..8, [1/2, 1/2, 1, 1, 1/2, 3/4, 1, 3/4, 3/4]);
+
+ points_x[3] := array(0..8, [0, 1/2, 1/2, 0, 1/4, 1/2, 1/4, 0, 1/4]);
+ points_y[3] := array(0..8, [1/2, 1/2, 1, 1, 1/2, 3/4, 1, 3/4, 3/4]);
+
+ prolongation := array(0..3,0..8, 0..8);
+
+ for i from 0 to 3 do
+ for j from 0 to 8 do
+ for k from 0 to 8 do
+ prolongation[i,j,k] := phi[k](points_x[i][j], points_y[i][j]);
+ od;
+ od;
+ od;
+
+ readlib(C);
+ C(prolongation);
+*/
+
+ prolongation[0](0,0) = 1.0;
+ prolongation[0](0,1) = 0.0;
+ prolongation[0](0,2) = 0.0;
+ prolongation[0](0,3) = 0.0;
+ prolongation[0](0,4) = 0.0;
+ prolongation[0](0,5) = 0.0;
+ prolongation[0](0,6) = 0.0;
+ prolongation[0](0,7) = 0.0;
+ prolongation[0](0,8) = 0.0;
+ prolongation[0](1,0) = 0.0;
+ prolongation[0](1,1) = 0.0;
+ prolongation[0](1,2) = 0.0;
+ prolongation[0](1,3) = 0.0;
+ prolongation[0](1,4) = 1.0;
+ prolongation[0](1,5) = 0.0;
+ prolongation[0](1,6) = 0.0;
+ prolongation[0](1,7) = 0.0;
+ prolongation[0](1,8) = 0.0;
+ prolongation[0](2,0) = 0.0;
+ prolongation[0](2,1) = 0.0;
+ prolongation[0](2,2) = 0.0;
+ prolongation[0](2,3) = 0.0;
+ prolongation[0](2,4) = 0.0;
+ prolongation[0](2,5) = 0.0;
+ prolongation[0](2,6) = 0.0;
+ prolongation[0](2,7) = 0.0;
+ prolongation[0](2,8) = 1.0;
+ prolongation[0](3,0) = 0.0;
+ prolongation[0](3,1) = 0.0;
+ prolongation[0](3,2) = 0.0;
+ prolongation[0](3,3) = 0.0;
+ prolongation[0](3,4) = 0.0;
+ prolongation[0](3,5) = 0.0;
+ prolongation[0](3,6) = 0.0;
+ prolongation[0](3,7) = 1.0;
+ prolongation[0](3,8) = 0.0;
+ prolongation[0](4,0) = 3.0/8.0;
+ prolongation[0](4,1) = -1.0/8.0;
+ prolongation[0](4,2) = 0.0;
+ prolongation[0](4,3) = 0.0;
+ prolongation[0](4,4) = 3.0/4.0;
+ prolongation[0](4,5) = 0.0;
+ prolongation[0](4,6) = 0.0;
+ prolongation[0](4,7) = 0.0;
+ prolongation[0](4,8) = 0.0;
+ prolongation[0](5,0) = 0.0;
+ prolongation[0](5,1) = 0.0;
+ prolongation[0](5,2) = 0.0;
+ prolongation[0](5,3) = 0.0;
+ prolongation[0](5,4) = 3.0/8.0;
+ prolongation[0](5,5) = 0.0;
+ prolongation[0](5,6) = -1.0/8.0;
+ prolongation[0](5,7) = 0.0;
+ prolongation[0](5,8) = 3.0/4.0;
+ prolongation[0](6,0) = 0.0;
+ prolongation[0](6,1) = 0.0;
+ prolongation[0](6,2) = 0.0;
+ prolongation[0](6,3) = 0.0;
+ prolongation[0](6,4) = 0.0;
+ prolongation[0](6,5) = -1.0/8.0;
+ prolongation[0](6,6) = 0.0;
+ prolongation[0](6,7) = 3.0/8.0;
+ prolongation[0](6,8) = 3.0/4.0;
+ prolongation[0](7,0) = 3.0/8.0;
+ prolongation[0](7,1) = 0.0;
+ prolongation[0](7,2) = 0.0;
+ prolongation[0](7,3) = -1.0/8.0;
+ prolongation[0](7,4) = 0.0;
+ prolongation[0](7,5) = 0.0;
+ prolongation[0](7,6) = 0.0;
+ prolongation[0](7,7) = 3.0/4.0;
+ prolongation[0](7,8) = 0.0;
+ prolongation[0](8,0) = 9.0/64.0;
+ prolongation[0](8,1) = -3.0/64.0;
+ prolongation[0](8,2) = 1.0/64.0;
+ prolongation[0](8,3) = -3.0/64.0;
+ prolongation[0](8,4) = 9.0/32.0;
+ prolongation[0](8,5) = -3.0/32.0;
+ prolongation[0](8,6) = -3.0/32.0;
+ prolongation[0](8,7) = 9.0/32.0;
+ prolongation[0](8,8) = 9.0/16.0;
+ prolongation[1](0,0) = 0.0;
+ prolongation[1](0,1) = 0.0;
+ prolongation[1](0,2) = 0.0;
+ prolongation[1](0,3) = 0.0;
+ prolongation[1](0,4) = 1.0;
+ prolongation[1](0,5) = 0.0;
+ prolongation[1](0,6) = 0.0;
+ prolongation[1](0,7) = 0.0;
+ prolongation[1](0,8) = 0.0;
+ prolongation[1](1,0) = 0.0;
+ prolongation[1](1,1) = 1.0;
+ prolongation[1](1,2) = 0.0;
+ prolongation[1](1,3) = 0.0;
+ prolongation[1](1,4) = 0.0;
+ prolongation[1](1,5) = 0.0;
+ prolongation[1](1,6) = 0.0;
+ prolongation[1](1,7) = 0.0;
+ prolongation[1](1,8) = 0.0;
+ prolongation[1](2,0) = 0.0;
+ prolongation[1](2,1) = 0.0;
+ prolongation[1](2,2) = 0.0;
+ prolongation[1](2,3) = 0.0;
+ prolongation[1](2,4) = 0.0;
+ prolongation[1](2,5) = 1.0;
+ prolongation[1](2,6) = 0.0;
+ prolongation[1](2,7) = 0.0;
+ prolongation[1](2,8) = 0.0;
+ prolongation[1](3,0) = 0.0;
+ prolongation[1](3,1) = 0.0;
+ prolongation[1](3,2) = 0.0;
+ prolongation[1](3,3) = 0.0;
+ prolongation[1](3,4) = 0.0;
+ prolongation[1](3,5) = 0.0;
+ prolongation[1](3,6) = 0.0;
+ prolongation[1](3,7) = 0.0;
+ prolongation[1](3,8) = 1.0;
+ prolongation[1](4,0) = -1.0/8.0;
+ prolongation[1](4,1) = 3.0/8.0;
+ prolongation[1](4,2) = 0.0;
+ prolongation[1](4,3) = 0.0;
+ prolongation[1](4,4) = 3.0/4.0;
+ prolongation[1](4,5) = 0.0;
+ prolongation[1](4,6) = 0.0;
+ prolongation[1](4,7) = 0.0;
+ prolongation[1](4,8) = 0.0;
+ prolongation[1](5,0) = 0.0;
+ prolongation[1](5,1) = 3.0/8.0;
+ prolongation[1](5,2) = -1.0/8.0;
+ prolongation[1](5,3) = 0.0;
+ prolongation[1](5,4) = 0.0;
+ prolongation[1](5,5) = 3.0/4.0;
+ prolongation[1](5,6) = 0.0;
+ prolongation[1](5,7) = 0.0;
+ prolongation[1](5,8) = 0.0;
+ prolongation[1](6,0) = 0.0;
+ prolongation[1](6,1) = 0.0;
+ prolongation[1](6,2) = 0.0;
+ prolongation[1](6,3) = 0.0;
+ prolongation[1](6,4) = 0.0;
+ prolongation[1](6,5) = 3.0/8.0;
+ prolongation[1](6,6) = 0.0;
+ prolongation[1](6,7) = -1.0/8.0;
+ prolongation[1](6,8) = 3.0/4.0;
+ prolongation[1](7,0) = 0.0;
+ prolongation[1](7,1) = 0.0;
+ prolongation[1](7,2) = 0.0;
+ prolongation[1](7,3) = 0.0;
+ prolongation[1](7,4) = 3.0/8.0;
+ prolongation[1](7,5) = 0.0;
+ prolongation[1](7,6) = -1.0/8.0;
+ prolongation[1](7,7) = 0.0;
+ prolongation[1](7,8) = 3.0/4.0;
+ prolongation[1](8,0) = -3.0/64.0;
+ prolongation[1](8,1) = 9.0/64.0;
+ prolongation[1](8,2) = -3.0/64.0;
+ prolongation[1](8,3) = 1.0/64.0;
+ prolongation[1](8,4) = 9.0/32.0;
+ prolongation[1](8,5) = 9.0/32.0;
+ prolongation[1](8,6) = -3.0/32.0;
+ prolongation[1](8,7) = -3.0/32.0;
+ prolongation[1](8,8) = 9.0/16.0;
+ prolongation[2](0,0) = 0.0;
+ prolongation[2](0,1) = 0.0;
+ prolongation[2](0,2) = 0.0;
+ prolongation[2](0,3) = 0.0;
+ prolongation[2](0,4) = 0.0;
+ prolongation[2](0,5) = 0.0;
+ prolongation[2](0,6) = 0.0;
+ prolongation[2](0,7) = 0.0;
+ prolongation[2](0,8) = 1.0;
+ prolongation[2](1,0) = 0.0;
+ prolongation[2](1,1) = 0.0;
+ prolongation[2](1,2) = 0.0;
+ prolongation[2](1,3) = 0.0;
+ prolongation[2](1,4) = 0.0;
+ prolongation[2](1,5) = 1.0;
+ prolongation[2](1,6) = 0.0;
+ prolongation[2](1,7) = 0.0;
+ prolongation[2](1,8) = 0.0;
+ prolongation[2](2,0) = 0.0;
+ prolongation[2](2,1) = 0.0;
+ prolongation[2](2,2) = 1.0;
+ prolongation[2](2,3) = 0.0;
+ prolongation[2](2,4) = 0.0;
+ prolongation[2](2,5) = 0.0;
+ prolongation[2](2,6) = 0.0;
+ prolongation[2](2,7) = 0.0;
+ prolongation[2](2,8) = 0.0;
+ prolongation[2](3,0) = 0.0;
+ prolongation[2](3,1) = 0.0;
+ prolongation[2](3,2) = 0.0;
+ prolongation[2](3,3) = 0.0;
+ prolongation[2](3,4) = 0.0;
+ prolongation[2](3,5) = 0.0;
+ prolongation[2](3,6) = 1.0;
+ prolongation[2](3,7) = 0.0;
+ prolongation[2](3,8) = 0.0;
+ prolongation[2](4,0) = 0.0;
+ prolongation[2](4,1) = 0.0;
+ prolongation[2](4,2) = 0.0;
+ prolongation[2](4,3) = 0.0;
+ prolongation[2](4,4) = 0.0;
+ prolongation[2](4,5) = 3.0/8.0;
+ prolongation[2](4,6) = 0.0;
+ prolongation[2](4,7) = -1.0/8.0;
+ prolongation[2](4,8) = 3.0/4.0;
+ prolongation[2](5,0) = 0.0;
+ prolongation[2](5,1) = -1.0/8.0;
+ prolongation[2](5,2) = 3.0/8.0;
+ prolongation[2](5,3) = 0.0;
+ prolongation[2](5,4) = 0.0;
+ prolongation[2](5,5) = 3.0/4.0;
+ prolongation[2](5,6) = 0.0;
+ prolongation[2](5,7) = 0.0;
+ prolongation[2](5,8) = 0.0;
+ prolongation[2](6,0) = 0.0;
+ prolongation[2](6,1) = 0.0;
+ prolongation[2](6,2) = 3.0/8.0;
+ prolongation[2](6,3) = -1.0/8.0;
+ prolongation[2](6,4) = 0.0;
+ prolongation[2](6,5) = 0.0;
+ prolongation[2](6,6) = 3.0/4.0;
+ prolongation[2](6,7) = 0.0;
+ prolongation[2](6,8) = 0.0;
+ prolongation[2](7,0) = 0.0;
+ prolongation[2](7,1) = 0.0;
+ prolongation[2](7,2) = 0.0;
+ prolongation[2](7,3) = 0.0;
+ prolongation[2](7,4) = -1.0/8.0;
+ prolongation[2](7,5) = 0.0;
+ prolongation[2](7,6) = 3.0/8.0;
+ prolongation[2](7,7) = 0.0;
+ prolongation[2](7,8) = 3.0/4.0;
+ prolongation[2](8,0) = 1.0/64.0;
+ prolongation[2](8,1) = -3.0/64.0;
+ prolongation[2](8,2) = 9.0/64.0;
+ prolongation[2](8,3) = -3.0/64.0;
+ prolongation[2](8,4) = -3.0/32.0;
+ prolongation[2](8,5) = 9.0/32.0;
+ prolongation[2](8,6) = 9.0/32.0;
+ prolongation[2](8,7) = -3.0/32.0;
+ prolongation[2](8,8) = 9.0/16.0;
+ prolongation[3](0,0) = 0.0;
+ prolongation[3](0,1) = 0.0;
+ prolongation[3](0,2) = 0.0;
+ prolongation[3](0,3) = 0.0;
+ prolongation[3](0,4) = 0.0;
+ prolongation[3](0,5) = 0.0;
+ prolongation[3](0,6) = 0.0;
+ prolongation[3](0,7) = 1.0;
+ prolongation[3](0,8) = 0.0;
+ prolongation[3](1,0) = 0.0;
+ prolongation[3](1,1) = 0.0;
+ prolongation[3](1,2) = 0.0;
+ prolongation[3](1,3) = 0.0;
+ prolongation[3](1,4) = 0.0;
+ prolongation[3](1,5) = 0.0;
+ prolongation[3](1,6) = 0.0;
+ prolongation[3](1,7) = 0.0;
+ prolongation[3](1,8) = 1.0;
+ prolongation[3](2,0) = 0.0;
+ prolongation[3](2,1) = 0.0;
+ prolongation[3](2,2) = 0.0;
+ prolongation[3](2,3) = 0.0;
+ prolongation[3](2,4) = 0.0;
+ prolongation[3](2,5) = 0.0;
+ prolongation[3](2,6) = 1.0;
+ prolongation[3](2,7) = 0.0;
+ prolongation[3](2,8) = 0.0;
+ prolongation[3](3,0) = 0.0;
+ prolongation[3](3,1) = 0.0;
+ prolongation[3](3,2) = 0.0;
+ prolongation[3](3,3) = 1.0;
+ prolongation[3](3,4) = 0.0;
+ prolongation[3](3,5) = 0.0;
+ prolongation[3](3,6) = 0.0;
+ prolongation[3](3,7) = 0.0;
+ prolongation[3](3,8) = 0.0;
+ prolongation[3](4,0) = 0.0;
+ prolongation[3](4,1) = 0.0;
+ prolongation[3](4,2) = 0.0;
+ prolongation[3](4,3) = 0.0;
+ prolongation[3](4,4) = 0.0;
+ prolongation[3](4,5) = -1.0/8.0;
+ prolongation[3](4,6) = 0.0;
+ prolongation[3](4,7) = 3.0/8.0;
+ prolongation[3](4,8) = 3.0/4.0;
+ prolongation[3](5,0) = 0.0;
+ prolongation[3](5,1) = 0.0;
+ prolongation[3](5,2) = 0.0;
+ prolongation[3](5,3) = 0.0;
+ prolongation[3](5,4) = -1.0/8.0;
+ prolongation[3](5,5) = 0.0;
+ prolongation[3](5,6) = 3.0/8.0;
+ prolongation[3](5,7) = 0.0;
+ prolongation[3](5,8) = 3.0/4.0;
+ prolongation[3](6,0) = 0.0;
+ prolongation[3](6,1) = 0.0;
+ prolongation[3](6,2) = -1.0/8.0;
+ prolongation[3](6,3) = 3.0/8.0;
+ prolongation[3](6,4) = 0.0;
+ prolongation[3](6,5) = 0.0;
+ prolongation[3](6,6) = 3.0/4.0;
+ prolongation[3](6,7) = 0.0;
+ prolongation[3](6,8) = 0.0;
+ prolongation[3](7,0) = -1.0/8.0;
+ prolongation[3](7,1) = 0.0;
+ prolongation[3](7,2) = 0.0;
+ prolongation[3](7,3) = 3.0/8.0;
+ prolongation[3](7,4) = 0.0;
+ prolongation[3](7,5) = 0.0;
+ prolongation[3](7,6) = 0.0;
+ prolongation[3](7,7) = 3.0/4.0;
+ prolongation[3](7,8) = 0.0;
+ prolongation[3](8,0) = -3.0/64.0;
+ prolongation[3](8,1) = 1.0/64.0;
+ prolongation[3](8,2) = -3.0/64.0;
+ prolongation[3](8,3) = 9.0/64.0;
+ prolongation[3](8,4) = -3.0/32.0;
+ prolongation[3](8,5) = -3.0/32.0;
+ prolongation[3](8,6) = 9.0/32.0;
+ prolongation[3](8,7) = 9.0/32.0;
+ prolongation[3](8,8) = 9.0/16.0;
+};
+
+
+template <>
+double
+FECubicSub<2>::shape_value (const unsigned int i,
+ const Point<2> &p) const
+{
+ Assert (i<total_dofs, ExcInvalidIndex(i));
+
+ const double xi = p(0),
+ eta= p(1);
+ switch (i)
+ {
+ case 0: return (1-xi)*( 2*xi-1) * (1-eta)*( 2*eta-1);
+ case 1: return xi *(-2*xi+1) * (1-eta)*( 2*eta-1);
+ case 2: return xi *(-2*xi+1) * eta *(-2*eta+1);
+ case 3: return (1-xi)*( 2*xi-1) * eta *(-2*eta+1);
+ case 4: return 4 * (1-xi)*xi * (1-eta)*(1-2*eta);
+ case 5: return 4 * xi *(-1+2*xi) * (1-eta)*eta;
+ case 6: return 4 * (1-xi)*xi * eta *(-1+2*eta);
+ case 7: return 4 * (1-xi)*(1-2*xi) * (1-eta)*eta;
+ case 8: return 16 * xi*(1-xi) * eta*(1-eta);
+ };
+ return 0;
+};
+
+
+
+template <>
+inline
+double
+FECubicSub<2>::linear_shape_value (const unsigned int i,
+ const Point<2>& p) const
+{
+ Assert((i<4), ExcInvalidIndex(i));
+ switch (i)
+ {
+ case 0: return (1.-p(0)) * (1.-p(1));
+ case 1: return p(0) * (1.-p(1));
+ case 2: return p(0) * p(1);
+ case 3: return (1.-p(0)) * p(1);
+ }
+ return 0.;
+};
+
+
+
+template <>
+Point<2>
+FECubicSub<2>::shape_grad (const unsigned int i,
+ const Point<2> &p) const
+{
+ Assert (i<total_dofs, ExcInvalidIndex(i));
+
+ const double xi = p(0),
+ eta= p(1);
+ switch (i)
+ {
+ case 0: return Point<2>(-(2*xi-1)*(1-eta)*(2*eta-1)+2*(1-xi)*(1-eta)*(2*eta-1),
+ -(1-xi)*(2*xi-1)*(2*eta-1)+2*(1-xi)*(2*xi-1)*(1-eta));
+ case 1: return Point<2>((-2*xi+1)*(1-eta)*(2*eta-1)-2*xi*(1-eta)*(2*eta-1),
+ -xi*(-2*xi+1)*(2*eta-1)+2*xi*(-2*xi+1)*(1-eta));
+ case 2: return Point<2>((-2*xi+1)*eta*(-2*eta+1)-2*xi*eta*(-2*eta+1),
+ xi*(-2*xi+1)*(-2*eta+1)-2*xi*(-2*xi+1)*eta);
+ case 3: return Point<2>(-(2*xi-1)*eta*(-2*eta+1)+2*(1-xi)*eta*(-2*eta+1),
+ (1-xi)*(2*xi-1)*(-2*eta+1)-2*(1-xi)*(2*xi-1)*eta);
+ case 4: return Point<2>(-4*xi*(1-eta)*(-2*eta+1)+4*(1-xi)*(1-eta)*(-2*eta+1),
+ -4*(1-xi)*xi*(-2*eta+1)-8*(1-xi)*xi*(1-eta));
+ case 5: return Point<2>(4*(2*xi-1)*(1-eta)*eta+8*xi*(1-eta)*eta,
+ -4*xi*(2*xi-1)*eta+4*xi*(2*xi-1)*(1-eta));
+ case 6: return Point<2>(-4*xi*eta*(2*eta-1)+4*(1-xi)*eta*(2*eta-1),
+ 4*(1-xi)*xi*(2*eta-1)+8*(1-xi)*xi*eta);
+ case 7: return Point<2>(-4*(-2*xi+1)*(1-eta)*eta-8*(1-xi)*(1-eta)*eta,
+ -4*(1-xi)*(-2*xi+1)*eta+4*(1-xi)*(-2*xi+1)*(1-eta));
+ case 8: return Point<2>(16*(1-xi)*(1-eta)*eta-16*xi*eta*(1-eta),
+ 16*xi*(1-xi)*(1-eta)-16*(1-xi)*xi*eta);
+ };
+ return Point<2> ();
+};
+
+
+
+template <>
+inline
+Point<2>
+FECubicSub<2>::linear_shape_grad (const unsigned int i,
+ const Point<2>& p) const
+{
+ Assert((i<4), ExcInvalidIndex(i));
+ switch (i)
+ {
+ case 0: return Point<2> (p(1)-1., p(0)-1.);
+ case 1: return Point<2> (1.-p(1), -p(0));
+ case 2: return Point<2> (p(1), p(0));
+ case 3: return Point<2> (-p(1), 1.-p(0));
+ }
+ return Point<2> ();
+};
+
+
+
+template <>
+void FECubicSub<2>::get_local_mass_matrix (const DoFHandler<2>::cell_iterator &cell,
+ const Boundary<2> &,
+ dFMatrix &local_mass_matrix) const {
+ Assert (local_mass_matrix.n() == total_dofs,
+ ExcWrongFieldDimension(local_mass_matrix.n(),total_dofs));
+ Assert (local_mass_matrix.m() == total_dofs,
+ ExcWrongFieldDimension(local_mass_matrix.m(),total_dofs));
+
+/* Get the computation of the local mass matrix by these lines in maple. Note
+ that tphi[i] are the basis function of the linear finite element, which
+ are used by the transformation (therefore >t<phi), while the phi[i]
+ are the basis functions of the biquadratic element.
+
+ x_real := sum(x[i]*tphi[i], i=0..3);
+ y_real := sum(y[i]*tphi[i], i=0..3);
+ tphi[0] := (1-xi)*(1-eta);
+ tphi[1] := xi*(1-eta);
+ tphi[2] := xi*eta;
+ tphi[3] := (1-xi)*eta;
+ detJ := diff(x_real,xi)*diff(y_real,eta) - diff(x_real,eta)*diff(y_real,xi);
+
+ phi[0] := (1-xi)*( 2*xi-1) * (1-eta)*( 2*eta-1);
+ phi[1] := xi *(-2*xi+1) * (1-eta)*( 2*eta-1);
+ phi[2] := xi *(-2*xi+1) * eta *(-2*eta+1);
+ phi[3] := (1-xi)*( 2*xi-1) * eta *(-2*eta+1);
+ phi[4] := 4 * (1-xi)*xi * (1-eta)*(1-2*eta);
+ phi[5] := 4 * xi *(-1+2*xi) * (1-eta)*eta;
+ phi[6] := 4 * (1-xi)*xi * eta *(-1+2*eta);
+ phi[7] := 4 * (1-xi)*(1-2*xi) * (1-eta)*eta;
+ phi[8] := 16 * xi*(1-xi) * eta*(1-eta);
+ m := proc (i,j) int( int(phi[i]*phi[j]*detJ, xi=0..1), eta=0..1); end;
+
+ M := array(0..8,0..8);
+ for i from 0 to 8 do
+ for j from 0 to 8 do
+ M[i,j] := m(i,j);
+ od;
+ od;
+
+ readlib(C);
+ C(M, optimized);
+*/
+
+ const double x[4] = { cell->vertex(0)(0),
+ cell->vertex(1)(0),
+ cell->vertex(2)(0),
+ cell->vertex(3)(0) };
+ const double y[4] = { cell->vertex(0)(1),
+ cell->vertex(1)(1),
+ cell->vertex(2)(1),
+ cell->vertex(3)(1) };
+
+/* check that the Jacobi determinant
+
+ t0 = (-x[0]*(1.0-eta)+x[1]*(1.0-eta)+x[2]*eta-x[3]*eta) *
+ (-y[0]*(1.0-xi)-y[1]*xi+y[2]*xi+y[3]*(1.0-xi)) -
+ (-x[0]*(1.0-xi)-x[1]*xi+x[2]*xi+x[3]*(1.0-xi)) *
+ (-y[0]*(1.0-eta)+y[1]*(1.0-eta)+y[2]*eta-y[3]*eta)
+
+ has the right sign.
+
+ We do not attempt to check its (hopefully) positive sign at all points
+ on the unit cell, but we check that it is positive in the four corners,
+ which is sufficient since $det J$ is a bilinear function.
+*/
+ Assert ((-x[0]+x[1])*(-y[0]+y[3])-(-x[0]+x[3])*(-y[0]+y[1]), // xi=eta=0
+ ExcJacobiDeterminantHasWrongSign());
+ Assert ((x[2]-x[3])*(-y[0]+y[3])-(-x[0]+x[3])*(y[2]-y[3]), // xi=0, eta=1
+ ExcJacobiDeterminantHasWrongSign());
+ Assert ((x[2]-x[3])*(-y[1]+y[2])-(-x[1]+x[2])*(y[2]-y[3]), // xi=eta=1
+ ExcJacobiDeterminantHasWrongSign());
+ Assert ((-x[0]+x[1])*(-y[1]+y[2])-(-x[1]+x[2])*(-y[0]+y[1]), // xi=1, eta=0
+ ExcJacobiDeterminantHasWrongSign());
+
+ const double t1 = (x[1]*y[0]);
+ const double t2 = (x[1]*y[2]);
+ const double t3 = (x[0]*y[3]);
+ const double t4 = (x[3]*y[2]);
+ const double t5 = (x[2]*y[3]);
+ const double t6 = (x[0]*y[1]);
+ const double t7 = (x[3]*y[1]);
+ const double t8 = (x[3]*y[0]);
+ const double t9 = (x[2]*y[1]);
+ const double t10 = (x[1]*y[3]);
+ const double t12 = (x[0]*y[2]);
+ const double t13 = (x[2]*y[0]);
+ const double t14 = (7.0/1800.0*t1-t2/450+t3/450+t4/1800-t5/1800-
+ 7.0/1800.0*t6+t12/600+
+ t7/600-t8/450-t13/600+t9/450-t10/600);
+ const double t15 = (-t1/1800+t2/1800-t3/1800-t4/1800+t5/1800+
+ t6/1800+t8/1800-t9/1800);
+ const double t16 = (t1/450-t2/1800+7.0/1800.0*t3+t4/450-
+ t5/450-t6/450-t12/600+t7/600
+ -7.0/1800.0*t8+t13/600+t9/1800-t10/600);
+ const double t17 = (-7.0/900.0*t1-2.0/225.0*t3-t4/900+t5/900
+ +7.0/900.0*t6+t12/900-7.0/
+ 900.0*t7+2.0/225.0*t8-t13/900+7.0/900.0*t10);
+ const double t18 = (t1/450-t2/900+t3/900-t6/450+t12/900+
+ t7/900-t8/900-t13/900+t9/900-
+ t10/900);
+ const double t19 = (t1/900+t3/450+t4/900-t5/900-t6/900
+ -t12/900+t7/900-t8/450+t13/900-
+ t10/900);
+ const double t20 = (-2.0/225.0*t1+t2/900-7.0/900.0*t3+
+ 2.0/225.0*t6-t12/900-7.0/900.0*t7
+ +7.0/900.0*t8+t13/900-t9/900+7.0/900.0*t10);
+ const double t21 = (-t1/225-t3/225+t6/225-t7/225+t8/225+t10/225);
+ const double t23 = (t1/450-7.0/1800.0*t2+t3/1800+t4/450
+ -t5/450-t6/450+t12/600-t7/600-t8
+ /1800-t13/600+7.0/1800.0*t9+t10/600);
+ const double t24 = (-7.0/900.0*t1+2.0/225.0*t2-t4/900+t5/900
+ +7.0/900.0*t6-7.0/900.0*t12
+ +t7/900+7.0/900.0*t13-2.0/225.0*t9-t10/900);
+ const double t25 = (-2.0/225.0*t1+7.0/900.0*t2-t3/900+2.0/225.0*t6
+ -7.0/900.0*t12-t7/900
+ +t8/900+7.0/900.0*t13-7.0/900.0*t9+t10/900);
+ const double t26 = (t1/900-t2/450+t4/900-t5/900-t6/900+t12/900
+ -t7/900-t13/900+t9/450+
+ t10/900);
+ const double t27 = (-t1/225+t2/225+t6/225-t12/225+t13/225-t9/225);
+ const double t29 = (t1/1800-t2/450+t3/450+7.0/1800.0*t4-7.0/1800.0*t5
+ -t6/1800-t12/600-
+ t7/600-t8/450+t13/600+t9/450+t10/600);
+ const double t30 = (7.0/900.0*t2-t3/900-2.0/225.0*t4+2.0/225.0*t5
+ +t12/900+7.0/900.0*t7+
+ t8/900-t13/900-7.0/900.0*t9-7.0/900.0*t10);
+ const double t31 = (-t1/900+2.0/225.0*t2-7.0/900.0*t4+7.0/900.0*t5
+ +t6/900-t12/900+7.0/
+ 900.0*t7+t13/900-2.0/225.0*t9-7.0/900.0*t10);
+ const double t32 = (-t2/900+t3/900+t4/450-t5/450-t12/900-t7/900
+ -t8/900+t13/900+t9/900+
+ t10/900);
+ const double t33 = (t2/225-t4/225+t5/225+t7/225-t9/225-t10/225);
+ const double t35 = (-t1/900-2.0/225.0*t3-7.0/900.0*t4+7.0/900.0*t5
+ +t6/900+7.0/900.0*t12
+ -t7/900+2.0/225.0*t8-7.0/900.0*t13+t10/900);
+ const double t36 = (t2/900-7.0/900.0*t3-2.0/225.0*t4+2.0/225.0*t5
+ +7.0/900.0*t12+t7/900+
+ 7.0/900.0*t8-7.0/900.0*t13-t9/900-t10/900);
+ const double t37 = (-t3/225-t4/225+t5/225+t12/225+t8/225-t13/225);
+ const double t38 = (-14.0/225.0*t1+8.0/225.0*t2-8.0/225.0*t3
+ -2.0/225.0*t4+2.0/225.0*t5+
+ 14.0/225.0*t6-2.0/75.0*t12-2.0/75.0*t7
+ +8.0/225.0*t8+2.0/75.0*t13-8.0/225.0*t9+
+ 2.0/75.0*t10);
+ const double t39 = (2.0/225.0*t1-2.0/225.0*t2+2.0/225.0*t3
+ +2.0/225.0*t4-2.0/225.0*t5
+ -2.0/225.0*t6-2.0/225.0*t8+2.0/225.0*t9);
+ const double t40 = (-8.0/225.0*t1+4.0/225.0*t2-4.0/225.0*t3
+ +8.0/225.0*t6-4.0/225.0*t12
+ -4.0/225.0*t7+4.0/225.0*t8+4.0/225.0*t13
+ -4.0/225.0*t9+4.0/225.0*t10);
+ const double t41 = (-8.0/225.0*t1+14.0/225.0*t2-2.0/225.0*t3
+ -8.0/225.0*t4+8.0/225.0*t5+
+ 8.0/225.0*t6-2.0/75.0*t12+2.0/75.0*t7
+ +2.0/225.0*t8+2.0/75.0*t13-14.0/225.0*t9
+ -2.0/75.0*t10);
+ const double t42 = (-4.0/225.0*t1+8.0/225.0*t2-4.0/225.0*t4
+ +4.0/225.0*t5+4.0/225.0*t6
+ -4.0/225.0*t12+4.0/225.0*t7+4.0/225.0*t13
+ -8.0/225.0*t9-4.0/225.0*t10);
+ const double t43 = (-2.0/225.0*t1+8.0/225.0*t2-8.0/225.0*t3
+ -14.0/225.0*t4+14.0/225.0*t5
+ +2.0/225.0*t6+2.0/75.0*t12+2.0/75.0*t7
+ +8.0/225.0*t8-2.0/75.0*t13-8.0/225.0*t9
+ -2.0/75.0*t10);
+ const double t44 = (4.0/225.0*t2-4.0/225.0*t3-8.0/225.0*t4
+ +8.0/225.0*t5+4.0/225.0*t12+
+ 4.0/225.0*t7+4.0/225.0*t8-4.0/225.0*t13
+ -4.0/225.0*t9-4.0/225.0*t10);
+ const double t45 = (-8.0/225.0*t1+2.0/225.0*t2-14.0/225.0*t3
+ -8.0/225.0*t4+8.0/225.0*t5+
+ 8.0/225.0*t6+2.0/75.0*t12-2.0/75.0*t7
+ +14.0/225.0*t8-2.0/75.0*t13-2.0/225.0*t9+
+ 2.0/75.0*t10);
+ const double t46 = (-4.0/225.0*t1-8.0/225.0*t3-4.0/225.0*t4
+ +4.0/225.0*t5+4.0/225.0*t6+
+ 4.0/225.0*t12-4.0/225.0*t7+8.0/225.0*t8
+ -4.0/225.0*t13+4.0/225.0*t10);
+
+ local_mass_matrix(0,0) = (-7.0/450.0*t1+t2/450-7.0/450.0*t3
+ -t4/450+t5/450+7.0/450.0*t6-t7/75
+ +7.0/450.0*t8-t9/450+t10/75);
+ local_mass_matrix(0,1) = (t14);
+ local_mass_matrix(0,2) = (t15);
+ local_mass_matrix(0,3) = (t16);
+ local_mass_matrix(0,4) = (t17);
+ local_mass_matrix(0,5) = (t18);
+ local_mass_matrix(0,6) = (t19);
+ local_mass_matrix(0,7) = (t20);
+ local_mass_matrix(0,8) = (t21);
+ local_mass_matrix(1,0) = (t14);
+ local_mass_matrix(1,1) = (-7.0/450.0*t1+7.0/450.0*t2-t3/450
+ -t4/450+t5/450+7.0/450.0*t6-
+ t12/75+t8/450+t13/75-7.0/450.0*t9);
+ local_mass_matrix(1,2) = (t23);
+ local_mass_matrix(1,3) = (t15);
+ local_mass_matrix(1,4) = (t24);
+ local_mass_matrix(1,5) = (t25);
+ local_mass_matrix(1,6) = (t26);
+ local_mass_matrix(1,7) = (t18);
+ local_mass_matrix(1,8) = (t27);
+ local_mass_matrix(2,0) = (t15);
+ local_mass_matrix(2,1) = (t23);
+ local_mass_matrix(2,2) = (-t1/450+7.0/450.0*t2-t3/450-7.0/450.0*t4
+ +7.0/450.0*t5+t6/450+t7/75
+ +t8/450-7.0/450.0*t9-t10/75);
+ local_mass_matrix(2,3) = (t29);
+ local_mass_matrix(2,4) = (t26);
+ local_mass_matrix(2,5) = (t30);
+ local_mass_matrix(2,6) = (t31);
+ local_mass_matrix(2,7) = (t32);
+ local_mass_matrix(2,8) = (t33);
+ local_mass_matrix(3,0) = (t16);
+ local_mass_matrix(3,1) = (t15);
+ local_mass_matrix(3,2) = (t29);
+ local_mass_matrix(3,3) = (-t1/450+t2/450-7.0/450.0*t3-7.0/450.0*t4
+ +7.0/450.0*t5+t6/450+
+ t12/75+7.0/450.0*t8-t13/75-t9/450);
+ local_mass_matrix(3,4) = (t19);
+ local_mass_matrix(3,5) = (t32);
+ local_mass_matrix(3,6) = (t35);
+ local_mass_matrix(3,7) = (t36);
+ local_mass_matrix(3,8) = (t37);
+ local_mass_matrix(4,0) = (t17);
+ local_mass_matrix(4,1) = (t24);
+ local_mass_matrix(4,2) = (t26);
+ local_mass_matrix(4,3) = (t19);
+ local_mass_matrix(4,4) = (t38);
+ local_mass_matrix(4,5) = (t27);
+ local_mass_matrix(4,6) = (t39);
+ local_mass_matrix(4,7) = (t21);
+ local_mass_matrix(4,8) = (t40);
+ local_mass_matrix(5,0) = (t18);
+ local_mass_matrix(5,1) = (t25);
+ local_mass_matrix(5,2) = (t30);
+ local_mass_matrix(5,3) = (t32);
+ local_mass_matrix(5,4) = (t27);
+ local_mass_matrix(5,5) = (t41);
+ local_mass_matrix(5,6) = (t33);
+ local_mass_matrix(5,7) = (t39);
+ local_mass_matrix(5,8) = (t42);
+ local_mass_matrix(6,0) = (t19);
+ local_mass_matrix(6,1) = (t26);
+ local_mass_matrix(6,2) = (t31);
+ local_mass_matrix(6,3) = (t35);
+ local_mass_matrix(6,4) = (t39);
+ local_mass_matrix(6,5) = (t33);
+ local_mass_matrix(6,6) = (t43);
+ local_mass_matrix(6,7) = (t37);
+ local_mass_matrix(6,8) = (t44);
+ local_mass_matrix(7,0) = (t20);
+ local_mass_matrix(7,1) = (t18);
+ local_mass_matrix(7,2) = (t32);
+ local_mass_matrix(7,3) = (t36);
+ local_mass_matrix(7,4) = (t21);
+ local_mass_matrix(7,5) = (t39);
+ local_mass_matrix(7,6) = (t37);
+ local_mass_matrix(7,7) = (t45);
+ local_mass_matrix(7,8) = (t46);
+ local_mass_matrix(8,0) = (t21);
+ local_mass_matrix(8,1) = (t27);
+ local_mass_matrix(8,2) = (t33);
+ local_mass_matrix(8,3) = (t37);
+ local_mass_matrix(8,4) = (t40);
+ local_mass_matrix(8,5) = (t42);
+ local_mass_matrix(8,6) = (t44);
+ local_mass_matrix(8,7) = (t46);
+ local_mass_matrix(8,8) = (-32.0/225.0*t1+32.0/225.0*t2-32.0/225.0*t3
+ -32.0/225.0*t4+32.0/225.0*t5+32.0/225.0*t6
+ +32.0/225.0*t8-32.0/225.0*t9);
+};
+
+
+
+template <>
+void FECubicSub<2>::get_ansatz_points (const typename DoFHandler<2>::cell_iterator &cell,
+ const Boundary<2>&,
+ vector<Point<2> > &ansatz_points) const {
+ Assert (ansatz_points.size() == total_dofs,
+ ExcWrongFieldDimension (ansatz_points.size(), total_dofs));
+
+ for (unsigned int vertex=0; vertex<4; ++vertex)
+ ansatz_points[vertex] = cell->vertex(vertex);
+
+ // for the bilinear mapping, the centers
+ // of the face on the unit cell are mapped
+ // to the mean coordinates of the vertices
+ for (unsigned int line=0; line<4; ++line)
+ ansatz_points[4+line] = (cell->line(line)->vertex(0) +
+ cell->line(line)->vertex(1)) / 2;
+ // same for the center of the square:
+ // since all four linear basis functions
+ // take on the value 1/4 at the center,
+ // the center is mapped to the mean
+ // coordinates of the four vertices
+ ansatz_points[8] = (ansatz_points[0] +
+ ansatz_points[1] +
+ ansatz_points[2] +
+ ansatz_points[3]) / 4;
+};
+
+
+
+template <>
+void FECubicSub<2>::get_face_ansatz_points (const typename DoFHandler<2>::face_iterator &face,
+ const Boundary<2> &,
+ vector<Point<2> > &ansatz_points) const {
+ Assert (ansatz_points.size() == dofs_per_face,
+ ExcWrongFieldDimension (ansatz_points.size(), dofs_per_face));
+
+ for (unsigned int vertex=0; vertex<2; ++vertex)
+ ansatz_points[vertex] = face->vertex(vertex);
+ ansatz_points[2] = (ansatz_points[0] + ansatz_points[1]) / 2;
+};
+
+
+
+template <>
+void FECubicSub<2>::get_face_jacobians (const DoFHandler<2>::face_iterator &face,
+ const Boundary<2> &,
+ const vector<Point<1> > &unit_points,
+ vector<double> &face_jacobians) const {
+ // more or less copied from the linear
+ // finite element
+ Assert (unit_points.size() == face_jacobians.size(),
+ ExcWrongFieldDimension (unit_points.size(), face_jacobians.size()));
+
+ // a linear mapping for a single line
+ // produces particularly simple
+ // expressions for the jacobi
+ // determinant :-)
+ const double h = sqrt((face->vertex(1) - face->vertex(0)).square());
+ fill_n (face_jacobians.begin(),
+ unit_points.size(),
+ h);
+};
+
+
+
+template <>
+void FECubicSub<2>::get_subface_jacobians (const DoFHandler<2>::face_iterator &face,
+ const unsigned int ,
+ const vector<Point<1> > &unit_points,
+ vector<double> &face_jacobians) const {
+ // more or less copied from the linear
+ // finite element
+ Assert (unit_points.size() == face_jacobians.size(),
+ ExcWrongFieldDimension (unit_points.size(), face_jacobians.size()));
+ Assert (face->at_boundary() == false,
+ ExcBoundaryFaceUsed ());
+
+ // a linear mapping for a single line
+ // produces particularly simple
+ // expressions for the jacobi
+ // determinant :-)
+ const double h = sqrt((face->vertex(1) - face->vertex(0)).square());
+ fill_n (face_jacobians.begin(),
+ unit_points.size(),
+ h/2);
+};
+
+
+
+template <>
+void FECubicSub<2>::get_normal_vectors (const DoFHandler<2>::cell_iterator &cell,
+ const unsigned int face_no,
+ const Boundary<2> &,
+ const vector<Point<1> > &unit_points,
+ vector<Point<2> > &normal_vectors) const {
+ // more or less copied from the linear
+ // finite element
+ Assert (unit_points.size() == normal_vectors.size(),
+ ExcWrongFieldDimension (unit_points.size(), normal_vectors.size()));
+
+ const DoFHandler<2>::face_iterator face = cell->face(face_no);
+ // compute direction of line
+ const Point<2> line_direction = (face->vertex(1) - face->vertex(0));
+ // rotate to the right by 90 degrees
+ const Point<2> normal_direction(line_direction(1),
+ -line_direction(0));
+
+ if (face_no <= 1)
+ // for sides 0 and 1: return the correctly
+ // scaled vector
+ fill (normal_vectors.begin(), normal_vectors.end(),
+ normal_direction / sqrt(normal_direction.square()));
+ else
+ // for sides 2 and 3: scale and invert
+ // vector
+ fill (normal_vectors.begin(), normal_vectors.end(),
+ normal_direction / (-sqrt(normal_direction.square())));
+};
+
+
+
+template <>
+void FECubicSub<2>::get_normal_vectors (const DoFHandler<2>::cell_iterator &cell,
+ const unsigned int face_no,
+ const unsigned int,
+ const vector<Point<1> > &unit_points,
+ vector<Point<2> > &normal_vectors) const {
+ // more or less copied from the linear
+ // finite element
+ // note, that in 2D the normal vectors to the
+ // subface have the same direction as that
+ // for the face
+ Assert (unit_points.size() == normal_vectors.size(),
+ ExcWrongFieldDimension (unit_points.size(), normal_vectors.size()));
+ Assert (cell->face(face_no)->at_boundary() == false,
+ ExcBoundaryFaceUsed ());
+
+ const DoFHandler<2>::face_iterator face = cell->face(face_no);
+ // compute direction of line
+ const Point<2> line_direction = (face->vertex(1) - face->vertex(0));
+ // rotate to the right by 90 degrees
+ const Point<2> normal_direction(line_direction(1),
+ -line_direction(0));
+
+ if (face_no <= 1)
+ // for sides 0 and 1: return the correctly
+ // scaled vector
+ fill (normal_vectors.begin(), normal_vectors.end(),
+ normal_direction / sqrt(normal_direction.square()));
+ else
+ // for sides 2 and 3: scale and invert
+ // vector
+ fill (normal_vectors.begin(), normal_vectors.end(),
+ normal_direction / (-sqrt(normal_direction.square())));
+};
+
+#endif
+
+
+
+
+
+template <int dim>
+void FECubicSub<dim>::fill_fe_values (const DoFHandler<dim>::cell_iterator &cell,
+ const vector<Point<dim> > &unit_points,
+ vector<dFMatrix> &jacobians,
+ const bool compute_jacobians,
+ vector<Point<dim> > &ansatz_points,
+ const bool compute_ansatz_points,
+ vector<Point<dim> > &q_points,
+ const bool compute_q_points,
+ const Boundary<dim> &boundary) const {
+ Assert (jacobians.size() == unit_points.size(),
+ ExcWrongFieldDimension(jacobians.size(), unit_points.size()));
+ Assert (q_points.size() == unit_points.size(),
+ ExcWrongFieldDimension(q_points.size(), unit_points.size()));
+ Assert (ansatz_points.size() == total_dofs,
+ ExcWrongFieldDimension(ansatz_points.size(), total_dofs));
+
+
+ unsigned int n_points=unit_points.size();
+
+ Point<dim> vertices[GeometryInfo<dim>::vertices_per_cell];
+ for (unsigned int l=0; l<GeometryInfo<dim>::vertices_per_cell; ++l)
+ vertices[l] = cell->vertex(l);
+
+
+ if (compute_q_points)
+ {
+ // initialize points to zero
+ for (unsigned int i=0; i<n_points; ++i)
+ q_points[i] = Point<dim> ();
+
+ // note: let x_l be the vector of the
+ // lth quadrature point in real space and
+ // xi_l that on the unit cell, let further
+ // p_j be the vector of the jth vertex
+ // of the cell in real space and
+ // N_j(xi_l) be the value of the associated
+ // basis function at xi_l, then
+ // x_l(xi_l) = sum_j p_j N_j(xi_l)
+ //
+ // Here, N_j is the *linear* basis function,
+ // not that of the finite element, since we
+ // use a subparametric mapping
+ for (unsigned int j=0; j<GeometryInfo<dim>::vertices_per_cell; ++j)
+ for (unsigned int l=0; l<n_points; ++l)
+ q_points[l] += vertices[j] * linear_shape_value(j, unit_points[l]);
+ };
+
+
+/* jacobi matrices: compute d(x)/d(xi) and invert this
+ Let M(l) be the inverse of J at the quadrature point l, then
+ M_{ij}(l) = sum_s p_i(s) d(N_s(l))/d(xi_j)
+ where p_i(s) is the i-th coordinate of the s-th vertex vector,
+ N_s(l) is the value of the s-th vertex shape function at the
+ quadrature point l.
+
+ We could therefore write:
+ l=0..n_points-1
+ i=0..dim-1
+ j=0..dim-1
+ M_{ij}(l) = 0
+ s=0..n_vertices
+ M_{ij}(l) += p_i(s) d(N_s(l))/d(xi_j)
+
+ However, we rewrite the loops to only compute the gradient once for
+ each integration point and basis function.
+*/
+ if (compute_jacobians)
+ {
+ dFMatrix M(dim,dim);
+ for (unsigned int l=0; l<n_points; ++l)
+ {
+ M.clear ();
+ for (unsigned int s=0; s<GeometryInfo<dim>::vertices_per_cell; ++s)
+ {
+ // we want the linear transform,
+ // so use that function
+ const Point<dim> gradient = linear_shape_grad (s, unit_points[l]);
+ for (unsigned int i=0; i<dim; ++i)
+ for (unsigned int j=0; j<dim; ++j)
+ M(i,j) += vertices[s](i) * gradient(j);
+ };
+ jacobians[l].invert(M);
+ };
+ };
+
+ // compute ansatz points, which are
+ // the corners for linear elements
+ if (compute_ansatz_points)
+ get_ansatz_points (cell, boundary, ansatz_points);
+};
+
+
+
+
+// explicit instantiations
+
+template class FECubicSub<deal_II_dimension>;
+