]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Some further changes for (bi-)quadratic elements.
authorwolf <wolf@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 25 Jun 1998 15:00:42 +0000 (15:00 +0000)
committerwolf <wolf@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 25 Jun 1998 15:00:42 +0000 (15:00 +0000)
git-svn-id: https://svn.dealii.org/trunk@412 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/deal.II/source/fe/fe_lib.linear.cc

index cfdf0bb20454fa99f5e510c064d91fad3fc32a8a..11ef7c4980d819348ca0299ad71f50b21f3d61dc 100644 (file)
@@ -766,6 +766,28 @@ void FEQuadraticSub<1>::get_normal_vectors (const DoFHandler<1>::cell_iterator &
   Assert (false, ExcInternalError());
 };
 
+
+
+template <>
+void FEQuadraticSub<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());
+
+  local_mass_matrix(0,0) = local_mass_matrix(1,1) = 2./15.*h;
+  local_mass_matrix(0,1) = local_mass_matrix(1,0) = -1./30.*h;
+  local_mass_matrix(0,2) = local_mass_matrix(2,0) = 1./15.*h;
+  local_mass_matrix(1,2) = local_mass_matrix(2,1) = 1./15.*h;
+  local_mass_matrix(2,2) = local_mass_matrix(2,2) = 8./15.*h;
+};
+
+
 #endif
 
 
@@ -800,14 +822,14 @@ FEQuadraticSub<2>::shape_value (const unsigned int i,
               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 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 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;
@@ -826,29 +848,306 @@ FEQuadraticSub<2>::shape_grad (const unsigned int i,
               eta= p(1);
   switch (i)
     {
-      case 0: return Point<2>((-4*xi+3) * (1.-eta)*( 2*eta-1),
-                             (1.-xi)*( 2*xi-1) * (-4*eta+3));
-      case 1: return Point<2>((-4*xi+1) * (1.-eta)*( 2*eta-1) ,
-                             xi *(-2*xi+1) * (-4*eta+3));
-      case 2: return Point<2>((-4*xi+1) *     eta *(-2*eta+1),
-                             xi *(-2*xi+1) * (-4*eta+1));
-      case 3: return Point<2>((-4*xi+3) *     eta *(-2*eta+1),
-                             (1.-xi)*( 2*xi-1) * (-4*eta+1));
-      case 4: return Point<2>(4 * (1-2*xi) * (1-eta)*(1-2*eta),
-                             4 * (1-xi)*xi * (4*eta-3));
-      case 5: return Point<2>(4 * (4*xi-1) * (1-eta)*eta,
-                             4 *    xi *(-1+2*xi) * (1-2*eta));
-      case 6: return Point<2>(4 * (1-2*xi) *    eta *(-1+2*eta),
-                             4 * (1-xi)*xi * (4*eta-1));
-      case 7: return Point<2>(4 * (4*xi-3) * (1-eta)*eta,
-                             4 * (1.-xi)*(1-2*xi) * (1-2*eta));
-      case 8: return Point<2>(16 * (1-2*xi) * eta*(1-eta),
-                             16 * xi*(1-xi) * (1-2*eta));
+      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 <>
+void FEQuadraticSub<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 FEQuadraticSub<2>::get_ansatz_points (const typename DoFHandler<2>::cell_iterator &cell,
                                           const Boundary<2>&,
@@ -1036,19 +1335,6 @@ void FEQuadraticSub<dim>::fill_fe_values (const DoFHandler<dim>::cell_iterator &
 
 
 
-
-template <int dim>
-void FEQuadraticSub<dim>::get_local_mass_matrix (const DoFHandler<dim>::cell_iterator &,
-                                             const Boundary<dim> &,
-                                             dFMatrix &) const {
-  Assert (false, ExcNotImplemented());
-};
-
-
-
-
-
-
 #if deal_II_dimension == 1
 
 template <>

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