Reference documentation for deal.II version Git 180c135554 2021-06-23 17:42:41 -0400
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fe_q_base.cc
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15 
16 
25 
26 #include <deal.II/fe/fe_dgp.h>
27 #include <deal.II/fe/fe_dgq.h>
28 #include <deal.II/fe/fe_nothing.h>
30 #include <deal.II/fe/fe_q_base.h>
33 #include <deal.II/fe/fe_tools.h>
34 #include <deal.II/fe/fe_wedge_p.h>
35 
36 #include <memory>
37 #include <sstream>
38 #include <vector>
39 
41 
42 
43 namespace internal
44 {
45  namespace FE_Q_Base
46  {
47  namespace
48  {
49  // in get_restriction_matrix() and get_prolongation_matrix(), want to undo
50  // tensorization on inner loops for performance reasons. this clears a
51  // dim-array
52  template <int dim>
53  inline void
54  zero_indices(unsigned int (&indices)[dim])
55  {
56  for (unsigned int d = 0; d < dim; ++d)
57  indices[d] = 0;
58  }
59 
60 
61 
62  // in get_restriction_matrix() and get_prolongation_matrix(), want to undo
63  // tensorization on inner loops for performance reasons. this increments
64  // tensor product indices
65  template <int dim>
66  inline void
67  increment_indices(unsigned int (&indices)[dim], const unsigned int dofs1d)
68  {
69  ++indices[0];
70  for (int d = 0; d < dim - 1; ++d)
71  if (indices[d] == dofs1d)
72  {
73  indices[d] = 0;
74  indices[d + 1]++;
75  }
76  }
77  } // namespace
78  } // namespace FE_Q_Base
79 } // namespace internal
80 
81 
82 
87 template <int xdim, int xspacedim>
88 struct FE_Q_Base<xdim, xspacedim>::Implementation
89 {
94  template <int spacedim>
95  static void
96  initialize_constraints(const std::vector<Point<1>> &,
98  {
99  // no constraints in 1d
100  }
101 
102 
103  template <int spacedim>
104  static void
105  initialize_constraints(const std::vector<Point<1>> & /*points*/,
107  {
108  const unsigned int dim = 2;
109 
110  unsigned int q_deg = fe.degree;
111  if (dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
112  &fe.get_poly_space()) != nullptr)
113  q_deg = fe.degree - 1;
114 
115  // restricted to each face, the traces of the shape functions is an
116  // element of P_{k} (in 2d), or Q_{k} (in 3d), where k is the degree of
117  // the element. from this, we interpolate between mother and cell face.
118 
119  // the interpolation process works as follows: on each subface, we want
120  // that finite element solutions from both sides coincide. i.e. if a and b
121  // are expansion coefficients for the shape functions from both sides, we
122  // seek a relation between a and b such that
123  // sum_j a_j phi^c_j(x) == sum_j b_j phi_j(x)
124  // for all points x on the interface. here, phi^c_j are the shape
125  // functions on the small cell on one side of the face, and phi_j those on
126  // the big cell on the other side. To get this relation, it suffices to
127  // look at a sufficient number of points for which this has to hold. if
128  // there are n functions, then we need n evaluation points, and we choose
129  // them equidistantly.
130  //
131  // we obtain the matrix system
132  // A a == B b
133  // where
134  // A_ij = phi^c_j(x_i)
135  // B_ij = phi_j(x_i)
136  // and the relation we are looking for is
137  // a = A^-1 B b
138  //
139  // for the special case of Lagrange interpolation polynomials, A_ij
140  // reduces to delta_ij, and
141  // a_i = B_ij b_j
142  // Hence, interface_constraints(i,j)=B_ij.
143  //
144  // for the general case, where we don't have Lagrange interpolation
145  // polynomials, this is a little more complicated. Then we would evaluate
146  // at a number of points and invert the interpolation matrix A.
147  //
148  // Note, that we build up these matrices for all subfaces at once, rather
149  // than considering them separately. the reason is that we finally will
150  // want to have them in this order anyway, as this is the format we need
151  // inside deal.II
152 
153  // In the following the points x_i are constructed in following order
154  // (n=degree-1)
155  // *----------*---------*
156  // 1..n 0 n+1..2n
157  // i.e. first the midpoint of the line, then the support points on subface
158  // 0 and on subface 1
159  std::vector<Point<dim - 1>> constraint_points;
160  // Add midpoint
161  constraint_points.emplace_back(0.5);
162 
163  if (q_deg > 1)
164  {
165  const unsigned int n = q_deg - 1;
166  const double step = 1. / q_deg;
167  // subface 0
168  for (unsigned int i = 1; i <= n; ++i)
169  constraint_points.push_back(
171  Point<dim - 1>(i * step), 0));
172  // subface 1
173  for (unsigned int i = 1; i <= n; ++i)
174  constraint_points.push_back(
176  Point<dim - 1>(i * step), 1));
177  }
178 
179  // Now construct relation between destination (child) and source (mother)
180  // dofs.
181 
182  fe.interface_constraints.TableBase<2, double>::reinit(
184 
185  // use that the element evaluates to 1 at index 0 and along the line at
186  // zero
187  const std::vector<unsigned int> &index_map_inverse =
189  const std::vector<unsigned int> face_index_map =
191  Assert(std::abs(
192  fe.poly_space->compute_value(index_map_inverse[0], Point<dim>()) -
193  1.) < 1e-14,
194  ExcInternalError());
195 
196  for (unsigned int i = 0; i < constraint_points.size(); ++i)
197  for (unsigned int j = 0; j < q_deg + 1; ++j)
198  {
199  Point<dim> p;
200  p[0] = constraint_points[i](0);
201  fe.interface_constraints(i, face_index_map[j]) =
202  fe.poly_space->compute_value(index_map_inverse[j], p);
203 
204  // if the value is small up to round-off, then simply set it to zero
205  // to avoid unwanted fill-in of the constraint matrices (which would
206  // then increase the number of other DoFs a constrained DoF would
207  // couple to)
208  if (std::fabs(fe.interface_constraints(i, face_index_map[j])) < 1e-13)
209  fe.interface_constraints(i, face_index_map[j]) = 0;
210  }
211  }
212 
213 
214  template <int spacedim>
215  static void
216  initialize_constraints(const std::vector<Point<1>> & /*points*/,
218  {
219  const unsigned int dim = 3;
220 
221  unsigned int q_deg = fe.degree;
222  if (dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
223  &fe.get_poly_space()) != nullptr)
224  q_deg = fe.degree - 1;
225 
226  // For a detailed documentation of the interpolation see the
227  // FE_Q_Base<2>::initialize_constraints function.
228 
229  // In the following the points x_i are constructed in the order as
230  // described in the documentation of the FiniteElement class (fe_base.h),
231  // i.e.
232  // *--15--4--16--*
233  // | | |
234  // 10 19 6 20 12
235  // | | |
236  // 1--7---0--8---2
237  // | | |
238  // 9 17 5 18 11
239  // | | |
240  // *--13--3--14--*
241  std::vector<Point<dim - 1>> constraint_points;
242 
243  // Add midpoint
244  constraint_points.emplace_back(0.5, 0.5);
245 
246  // Add midpoints of lines of "mother-face"
247  constraint_points.emplace_back(0, 0.5);
248  constraint_points.emplace_back(1, 0.5);
249  constraint_points.emplace_back(0.5, 0);
250  constraint_points.emplace_back(0.5, 1);
251 
252  if (q_deg > 1)
253  {
254  const unsigned int n = q_deg - 1;
255  const double step = 1. / q_deg;
256  std::vector<Point<dim - 2>> line_support_points(n);
257  for (unsigned int i = 0; i < n; ++i)
258  line_support_points[i](0) = (i + 1) * step;
259  Quadrature<dim - 2> qline(line_support_points);
260 
261  // auxiliary points in 2d
262  std::vector<Point<dim - 1>> p_line(n);
263 
264  // Add nodes of lines interior in the "mother-face"
265 
266  // line 5: use line 9
268  ReferenceCells::get_hypercube<dim - 1>(), qline, 0, 0, p_line);
269  for (unsigned int i = 0; i < n; ++i)
270  constraint_points.push_back(p_line[i] + Point<dim - 1>(0.5, 0));
271  // line 6: use line 10
273  ReferenceCells::get_hypercube<dim - 1>(), qline, 0, 1, p_line);
274  for (unsigned int i = 0; i < n; ++i)
275  constraint_points.push_back(p_line[i] + Point<dim - 1>(0.5, 0));
276  // line 7: use line 13
278  ReferenceCells::get_hypercube<dim - 1>(), qline, 2, 0, p_line);
279  for (unsigned int i = 0; i < n; ++i)
280  constraint_points.push_back(p_line[i] + Point<dim - 1>(0, 0.5));
281  // line 8: use line 14
283  ReferenceCells::get_hypercube<dim - 1>(), qline, 2, 1, p_line);
284  for (unsigned int i = 0; i < n; ++i)
285  constraint_points.push_back(p_line[i] + Point<dim - 1>(0, 0.5));
286 
287  // DoFs on bordering lines lines 9-16
288  for (unsigned int face = 0;
289  face < GeometryInfo<dim - 1>::faces_per_cell;
290  ++face)
291  for (unsigned int subface = 0;
292  subface < GeometryInfo<dim - 1>::max_children_per_face;
293  ++subface)
294  {
296  ReferenceCells::get_hypercube<dim - 1>(),
297  qline,
298  face,
299  subface,
300  p_line);
301  constraint_points.insert(constraint_points.end(),
302  p_line.begin(),
303  p_line.end());
304  }
305 
306  // Create constraints for interior nodes
307  std::vector<Point<dim - 1>> inner_points(n * n);
308  for (unsigned int i = 0, iy = 1; iy <= n; ++iy)
309  for (unsigned int ix = 1; ix <= n; ++ix)
310  inner_points[i++] = Point<dim - 1>(ix * step, iy * step);
311 
312  // at the moment do this for isotropic face refinement only
313  for (unsigned int child = 0;
314  child < GeometryInfo<dim - 1>::max_children_per_cell;
315  ++child)
316  for (const auto &inner_point : inner_points)
317  constraint_points.push_back(
319  child));
320  }
321 
322  // Now construct relation between destination (child) and source (mother)
323  // dofs.
324  const unsigned int pnts = (q_deg + 1) * (q_deg + 1);
325 
326  // use that the element evaluates to 1 at index 0 and along the line at
327  // zero
328  const std::vector<unsigned int> &index_map_inverse =
330  const std::vector<unsigned int> face_index_map =
332  Assert(std::abs(
333  fe.poly_space->compute_value(index_map_inverse[0], Point<dim>()) -
334  1.) < 1e-14,
335  ExcInternalError());
336 
337  fe.interface_constraints.TableBase<2, double>::reinit(
339 
340  for (unsigned int i = 0; i < constraint_points.size(); ++i)
341  {
342  const double interval = static_cast<double>(q_deg * 2);
343  bool mirror[dim - 1];
344  Point<dim> constraint_point;
345 
346  // Eliminate FP errors in constraint points. Due to their origin, they
347  // must all be fractions of the unit interval. If we have polynomial
348  // degree 4, the refined element has 8 intervals. Hence the
349  // coordinates must be 0, 0.125, 0.25, 0.375 etc. Now the coordinates
350  // of the constraint points will be multiplied by the inverse of the
351  // interval size (in the example by 8). After that the coordinates
352  // must be integral numbers. Hence a normal truncation is performed
353  // and the coordinates will be scaled back. The equal treatment of all
354  // coordinates should eliminate any FP errors.
355  for (unsigned int k = 0; k < dim - 1; ++k)
356  {
357  const int coord_int =
358  static_cast<int>(constraint_points[i](k) * interval + 0.25);
359  constraint_point(k) = 1. * coord_int / interval;
360 
361  // The following lines of code should eliminate the problems with
362  // the constraints object which appeared for P>=4. The
363  // AffineConstraints class complained about different constraints
364  // for the same entry: Actually, this
365  // difference could be attributed to FP errors, as it was in the
366  // range of 1.0e-16. These errors originate in the loss of
367  // symmetry in the FP approximation of the shape-functions.
368  // Considering a 3rd order shape function in 1D, we have
369  // N0(x)=N3(1-x) and N1(x)=N2(1-x). For higher order polynomials
370  // the FP approximations of the shape functions do not satisfy
371  // these equations any more! Thus in the following code
372  // everything is computed in the interval x \in [0..0.5], which is
373  // sufficient to express all values that could come out from a
374  // computation of any shape function in the full interval
375  // [0..1]. If x > 0.5 the computation is done for 1-x with the
376  // shape function N_{p-n} instead of N_n. Hence symmetry is
377  // preserved and everything works fine...
378  //
379  // For a different explanation of the problem, see the discussion
380  // in the FiniteElement class for constraint matrices in 3d.
381  mirror[k] = (constraint_point(k) > 0.5);
382  if (mirror[k])
383  constraint_point(k) = 1.0 - constraint_point(k);
384  }
385 
386  for (unsigned int j = 0; j < pnts; ++j)
387  {
388  unsigned int indices[2] = {j % (q_deg + 1), j / (q_deg + 1)};
389 
390  for (unsigned int k = 0; k < 2; ++k)
391  if (mirror[k])
392  indices[k] = q_deg - indices[k];
393 
394  const unsigned int new_index =
395  indices[1] * (q_deg + 1) + indices[0];
396 
397  fe.interface_constraints(i, face_index_map[j]) =
398  fe.poly_space->compute_value(index_map_inverse[new_index],
399  constraint_point);
400 
401  // if the value is small up to round-off, then simply set it to
402  // zero to avoid unwanted fill-in of the constraint matrices
403  // (which would then increase the number of other DoFs a
404  // constrained DoF would couple to)
405  if (std::fabs(fe.interface_constraints(i, face_index_map[j])) <
406  1e-13)
407  fe.interface_constraints(i, face_index_map[j]) = 0;
408  }
409  }
410  }
411 };
412 
413 
414 
415 template <int dim, int spacedim>
417  const ScalarPolynomialsBase<dim> &poly_space,
418  const FiniteElementData<dim> & fe_data,
419  const std::vector<bool> & restriction_is_additive_flags)
420  : FE_Poly<dim, spacedim>(
421  poly_space,
422  fe_data,
423  restriction_is_additive_flags,
424  std::vector<ComponentMask>(1, std::vector<bool>(1, true)))
425  , q_degree(dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
426  &poly_space) != nullptr ?
427  this->degree - 1 :
428  this->degree)
429 {}
430 
431 
432 
433 template <int dim, int spacedim>
434 void
436 {
437  Assert(points[0][0] == 0,
438  ExcMessage("The first support point has to be zero."));
439  Assert(points.back()[0] == 1,
440  ExcMessage("The last support point has to be one."));
441 
442  // distinguish q/q_dg0 case: need to be flexible enough to allow more
443  // degrees of freedom than there are FE_Q degrees of freedom for derived
444  // class FE_Q_DG0 that otherwise shares 95% of the code.
445  const unsigned int q_dofs_per_cell =
446  Utilities::fixed_power<dim>(q_degree + 1);
447  Assert(q_dofs_per_cell == this->n_dofs_per_cell() ||
448  q_dofs_per_cell + 1 == this->n_dofs_per_cell() ||
449  q_dofs_per_cell + dim == this->n_dofs_per_cell(),
450  ExcInternalError());
451 
452  [this, q_dofs_per_cell]() {
453  std::vector<unsigned int> renumber =
454  FETools::hierarchic_to_lexicographic_numbering<dim>(q_degree);
455  for (unsigned int i = q_dofs_per_cell; i < this->n_dofs_per_cell(); ++i)
456  renumber.push_back(i);
457  auto *tensor_poly_space_ptr =
458  dynamic_cast<TensorProductPolynomials<dim> *>(this->poly_space.get());
459  if (tensor_poly_space_ptr != nullptr)
460  {
461  tensor_poly_space_ptr->set_numbering(renumber);
462  return;
463  }
464  auto *tensor_piecewise_poly_space_ptr = dynamic_cast<
466  *>(this->poly_space.get());
467  if (tensor_piecewise_poly_space_ptr != nullptr)
468  {
469  tensor_piecewise_poly_space_ptr->set_numbering(renumber);
470  return;
471  }
472  auto *tensor_bubbles_poly_space_ptr =
473  dynamic_cast<TensorProductPolynomialsBubbles<dim> *>(
474  this->poly_space.get());
475  if (tensor_bubbles_poly_space_ptr != nullptr)
476  {
477  tensor_bubbles_poly_space_ptr->set_numbering(renumber);
478  return;
479  }
480  auto *tensor_const_poly_space_ptr =
481  dynamic_cast<TensorProductPolynomialsConst<dim> *>(
482  this->poly_space.get());
483  if (tensor_const_poly_space_ptr != nullptr)
484  {
485  tensor_const_poly_space_ptr->set_numbering(renumber);
486  return;
487  }
488  Assert(false, ExcNotImplemented());
489  }();
490 
491  // Finally fill in support points on cell and face and initialize
492  // constraints. All of this can happen in parallel
493  Threads::TaskGroup<> tasks;
494  tasks += Threads::new_task([&]() { initialize_unit_support_points(points); });
495  tasks +=
497  tasks += Threads::new_task([&]() { initialize_constraints(points); });
498  tasks +=
500  tasks.join_all();
501 
502  // do not initialize embedding and restriction here. these matrices are
503  // initialized on demand in get_restriction_matrix and
504  // get_prolongation_matrix
505 }
506 
507 
508 
509 template <int dim, int spacedim>
510 void
512  const FiniteElement<dim, spacedim> &x_source_fe,
513  FullMatrix<double> & interpolation_matrix) const
514 {
515  // go through the list of elements we can interpolate from
516  if (const FE_Q_Base<dim, spacedim> *source_fe =
517  dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&x_source_fe))
518  {
519  // ok, source is a Q element, so we will be able to do the work
520  Assert(interpolation_matrix.m() == this->n_dofs_per_cell(),
521  ExcDimensionMismatch(interpolation_matrix.m(),
522  this->n_dofs_per_cell()));
523  Assert(interpolation_matrix.n() == x_source_fe.n_dofs_per_cell(),
524  ExcDimensionMismatch(interpolation_matrix.m(),
525  x_source_fe.n_dofs_per_cell()));
526 
527  // only evaluate Q dofs
528  const unsigned int q_dofs_per_cell =
529  Utilities::fixed_power<dim>(q_degree + 1);
530  const unsigned int source_q_dofs_per_cell =
531  Utilities::fixed_power<dim>(source_fe->degree + 1);
532 
533  // evaluation is simply done by evaluating the other FE's basis functions
534  // on the unit support points (FE_Q has the property that the cell
535  // interpolation matrix is a unit matrix, so no need to evaluate it and
536  // invert it)
537  for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
538  {
539  // read in a point on this cell and evaluate the shape functions there
540  const Point<dim> p = this->unit_support_points[j];
541 
542  // FE_Q element evaluates to 1 in unit support point and to zero in
543  // all other points by construction
544  Assert(std::abs(this->poly_space->compute_value(j, p) - 1.) < 1e-13,
545  ExcInternalError());
546 
547  for (unsigned int i = 0; i < source_q_dofs_per_cell; ++i)
548  interpolation_matrix(j, i) =
549  source_fe->poly_space->compute_value(i, p);
550  }
551 
552  // for FE_Q_DG0, add one last row of identity
553  if (q_dofs_per_cell < this->n_dofs_per_cell())
554  {
555  AssertDimension(source_q_dofs_per_cell + 1,
556  source_fe->n_dofs_per_cell());
557  for (unsigned int i = 0; i < source_q_dofs_per_cell; ++i)
558  interpolation_matrix(q_dofs_per_cell, i) = 0.;
559  for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
560  interpolation_matrix(j, source_q_dofs_per_cell) = 0.;
561  interpolation_matrix(q_dofs_per_cell, source_q_dofs_per_cell) = 1.;
562  }
563 
564  // cut off very small values
565  const double eps = 2e-13 * q_degree * dim;
566  for (unsigned int i = 0; i < this->n_dofs_per_cell(); ++i)
567  for (unsigned int j = 0; j < source_fe->n_dofs_per_cell(); ++j)
568  if (std::fabs(interpolation_matrix(i, j)) < eps)
569  interpolation_matrix(i, j) = 0.;
570 
571  // make sure that the row sum of each of the matrices is 1 at this
572  // point. this must be so since the shape functions sum up to 1
573  for (unsigned int i = 0; i < this->n_dofs_per_cell(); ++i)
574  {
575  double sum = 0.;
576  for (unsigned int j = 0; j < source_fe->n_dofs_per_cell(); ++j)
577  sum += interpolation_matrix(i, j);
578 
579  Assert(std::fabs(sum - 1) < eps, ExcInternalError());
580  }
581  }
582  else if (dynamic_cast<const FE_Nothing<dim> *>(&x_source_fe))
583  {
584  // the element we want to interpolate from is an FE_Nothing. this
585  // element represents a function that is constant zero and has no
586  // degrees of freedom, so the interpolation is simply a multiplication
587  // with a n_dofs x 0 matrix. there is nothing to do here
588 
589  // we would like to verify that the number of rows and columns of
590  // the matrix equals this->n_dofs_per_cell() and zero. unfortunately,
591  // whenever we do FullMatrix::reinit(m,0), it sets both rows and
592  // columns to zero, instead of m and zero. thus, only test the
593  // number of columns
594  Assert(interpolation_matrix.n() == x_source_fe.n_dofs_per_cell(),
595  ExcDimensionMismatch(interpolation_matrix.m(),
596  x_source_fe.n_dofs_per_cell()));
597  }
598  else
599  AssertThrow(
600  false,
601  (typename FiniteElement<dim,
602  spacedim>::ExcInterpolationNotImplemented()));
603 }
604 
605 
606 
607 template <int dim, int spacedim>
608 void
610  const FiniteElement<dim, spacedim> &source_fe,
611  FullMatrix<double> & interpolation_matrix,
612  const unsigned int face_no) const
613 {
616  interpolation_matrix,
617  face_no);
618 }
619 
620 
621 
622 template <int dim, int spacedim>
623 void
625  const FiniteElement<dim, spacedim> &source_fe,
626  const unsigned int subface,
627  FullMatrix<double> & interpolation_matrix,
628  const unsigned int face_no) const
629 {
630  Assert(interpolation_matrix.m() == source_fe.n_dofs_per_face(face_no),
631  ExcDimensionMismatch(interpolation_matrix.m(),
632  source_fe.n_dofs_per_face(face_no)));
633 
634  Assert(source_fe.n_components() == this->n_components(),
635  ExcDimensionMismatch(source_fe.n_components(), this->n_components()));
636 
637  if (source_fe.has_face_support_points(face_no))
638  {
639  // have this test in here since a table of size 2x0 reports its size as
640  // 0x0
641  Assert(interpolation_matrix.n() == this->n_dofs_per_face(face_no),
642  ExcDimensionMismatch(interpolation_matrix.n(),
643  this->n_dofs_per_face(face_no)));
644 
645  // Make sure that the element for which the DoFs should be constrained
646  // is the one with the higher polynomial degree. Actually the procedure
647  // will work also if this assertion is not satisfied. But the matrices
648  // produced in that case might lead to problems in the hp-procedures,
649  // which use this method.
650  Assert(
651  this->n_dofs_per_face(face_no) <= source_fe.n_dofs_per_face(face_no),
652  (typename FiniteElement<dim,
654 
655  // generate a point on this cell and evaluate the shape functions there
656  const Quadrature<dim - 1> quad_face_support(
657  source_fe.get_unit_face_support_points(face_no));
658 
659  // Rule of thumb for FP accuracy, that can be expected for a given
660  // polynomial degree. This value is used to cut off values close to
661  // zero.
662  const double eps = 2e-13 * this->q_degree * std::max(dim - 1, 1);
663 
664  // compute the interpolation matrix by simply taking the value at the
665  // support points.
666  // TODO: Verify that all faces are the same with respect to
667  // these support points. Furthermore, check if something has to
668  // be done for the face orientation flag in 3D.
669  const Quadrature<dim> subface_quadrature =
670  subface == numbers::invalid_unsigned_int ?
672  quad_face_support,
673  0) :
675  quad_face_support,
676  0,
677  subface);
678  for (unsigned int i = 0; i < source_fe.n_dofs_per_face(face_no); ++i)
679  {
680  const Point<dim> &p = subface_quadrature.point(i);
681 
682  for (unsigned int j = 0; j < this->n_dofs_per_face(face_no); ++j)
683  {
684  double matrix_entry =
685  this->shape_value(this->face_to_cell_index(j, 0), p);
686 
687  // Correct the interpolated value. I.e. if it is close to 1 or
688  // 0, make it exactly 1 or 0. Unfortunately, this is required to
689  // avoid problems with higher order elements.
690  if (std::fabs(matrix_entry - 1.0) < eps)
691  matrix_entry = 1.0;
692  if (std::fabs(matrix_entry) < eps)
693  matrix_entry = 0.0;
694 
695  interpolation_matrix(i, j) = matrix_entry;
696  }
697  }
698 
699 #ifdef DEBUG
700  // make sure that the row sum of each of the matrices is 1 at this
701  // point. this must be so since the shape functions sum up to 1
702  for (unsigned int j = 0; j < source_fe.n_dofs_per_face(face_no); ++j)
703  {
704  double sum = 0.;
705 
706  for (unsigned int i = 0; i < this->n_dofs_per_face(face_no); ++i)
707  sum += interpolation_matrix(j, i);
708 
709  Assert(std::fabs(sum - 1) < eps, ExcInternalError());
710  }
711 #endif
712  }
713  else if (dynamic_cast<const FE_Nothing<dim> *>(&source_fe) != nullptr)
714  {
715  // nothing to do here, the FE_Nothing has no degrees of freedom anyway
716  }
717  else
718  AssertThrow(
719  false,
720  (typename FiniteElement<dim,
721  spacedim>::ExcInterpolationNotImplemented()));
722 }
723 
724 
725 
726 template <int dim, int spacedim>
727 bool
729 {
730  return true;
731 }
732 
733 
734 
735 template <int dim, int spacedim>
736 std::vector<std::pair<unsigned int, unsigned int>>
738  const FiniteElement<dim, spacedim> &fe_other) const
739 {
740  if (dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other) != nullptr)
741  {
742  // there should be exactly one single DoF of each FE at a vertex, and they
743  // should have identical value
744  return {{0U, 0U}};
745  }
746  else if (dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other) !=
747  nullptr)
748  {
749  // there should be exactly one single DoF of each FE at a vertex, and they
750  // should have identical value
751  return {{0U, 0U}};
752  }
753  else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
754  {
755  // the FE_Nothing has no degrees of freedom, so there are no
756  // equivalencies to be recorded
757  return {};
758  }
759  else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
760  {
761  // if the other element has no elements on faces at all,
762  // then it would be impossible to enforce any kind of
763  // continuity even if we knew exactly what kind of element
764  // we have -- simply because the other element declares
765  // that it is discontinuous because it has no DoFs on
766  // its faces. in that case, just state that we have no
767  // constraints to declare
768  return {};
769  }
770  else
771  {
772  Assert(false, ExcNotImplemented());
773  return {};
774  }
775 }
776 
777 
778 
779 template <int dim, int spacedim>
780 std::vector<std::pair<unsigned int, unsigned int>>
782  const FiniteElement<dim, spacedim> &fe_other) const
783 {
784  // we can presently only compute these identities if both FEs are FE_Qs or
785  // if the other one is an FE_Nothing
786  if (const FE_Q_Base<dim, spacedim> *fe_q_other =
787  dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other))
788  {
789  // dofs are located along lines, so two dofs are identical if they are
790  // located at identical positions. if we had only equidistant points, we
791  // could simply check for similarity like (i+1)*q == (j+1)*p, but we
792  // might have other support points (e.g. Gauss-Lobatto
793  // points). Therefore, read the points in unit_support_points for the
794  // first coordinate direction. We take the lexicographic ordering of the
795  // points in the first direction (i.e., x-direction), which we access
796  // between index 1 and p-1 (index 0 and p are vertex dofs).
797  const unsigned int p = this->degree;
798  const unsigned int q = fe_q_other->degree;
799 
800  std::vector<std::pair<unsigned int, unsigned int>> identities;
801 
802  const std::vector<unsigned int> &index_map_inverse =
804  const std::vector<unsigned int> &index_map_inverse_other =
805  fe_q_other->get_poly_space_numbering_inverse();
806 
807  for (unsigned int i = 0; i < p - 1; ++i)
808  for (unsigned int j = 0; j < q - 1; ++j)
809  if (std::fabs(
810  this->unit_support_points[index_map_inverse[i + 1]][0] -
811  fe_q_other->unit_support_points[index_map_inverse_other[j + 1]]
812  [0]) < 1e-14)
813  identities.emplace_back(i, j);
814 
815  return identities;
816  }
817  else if (const FE_SimplexP<dim, spacedim> *fe_p_other =
818  dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other))
819  {
820  // DoFs are located along lines, so two dofs are identical if they are
821  // located at identical positions. If we had only equidistant points, we
822  // could simply check for similarity like (i+1)*q == (j+1)*p, but we
823  // might have other support points (e.g. Gauss-Lobatto
824  // points). Therefore, read the points in unit_support_points for the
825  // first coordinate direction. For FE_Q, we take the lexicographic
826  // ordering of the line support points in the first direction (i.e.,
827  // x-direction), which we access between index 1 and p-1 (index 0 and p
828  // are vertex dofs). For FE_SimplexP, they are currently hard-coded and we
829  // iterate over points on the first line which begin after the 3 vertex
830  // points in the complete list of unit support points
831 
832  Assert(fe_p_other->degree <= 2, ExcNotImplemented());
833 
834  const std::vector<unsigned int> &index_map_inverse_q =
836 
837  std::vector<std::pair<unsigned int, unsigned int>> identities;
838 
839  for (unsigned int i = 0; i < this->degree - 1; ++i)
840  for (unsigned int j = 0; j < fe_p_other->degree - 1; ++j)
841  if (std::fabs(
842  this->unit_support_points[index_map_inverse_q[i + 1]][0] -
843  fe_p_other->get_unit_support_points()[j + 3][0]) < 1e-14)
844  identities.emplace_back(i, j);
845 
846  return identities;
847  }
848  else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
849  {
850  // the FE_Nothing has no degrees of freedom, so there are no
851  // equivalencies to be recorded
852  return {};
853  }
854  else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
855  {
856  // if the other element has no elements on faces at all,
857  // then it would be impossible to enforce any kind of
858  // continuity even if we knew exactly what kind of element
859  // we have -- simply because the other element declares
860  // that it is discontinuous because it has no DoFs on
861  // its faces. in that case, just state that we have no
862  // constraints to declare
863  return {};
864  }
865  else
866  {
867  Assert(false, ExcNotImplemented());
868  return {};
869  }
870 }
871 
872 
873 
874 template <int dim, int spacedim>
875 std::vector<std::pair<unsigned int, unsigned int>>
877  const FiniteElement<dim, spacedim> &fe_other,
878  const unsigned int) const
879 {
880  // we can presently only compute these identities if both FEs are FE_Qs or
881  // if the other one is an FE_Nothing
882  if (const FE_Q_Base<dim, spacedim> *fe_q_other =
883  dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other))
884  {
885  // this works exactly like the line case above, except that now we have
886  // to have two indices i1, i2 and j1, j2 to characterize the dofs on the
887  // face of each of the finite elements. since they are ordered
888  // lexicographically along the first line and we have a tensor product,
889  // the rest is rather straightforward
890  const unsigned int p = this->degree;
891  const unsigned int q = fe_q_other->degree;
892 
893  std::vector<std::pair<unsigned int, unsigned int>> identities;
894 
895  const std::vector<unsigned int> &index_map_inverse =
897  const std::vector<unsigned int> &index_map_inverse_other =
898  fe_q_other->get_poly_space_numbering_inverse();
899 
900  for (unsigned int i1 = 0; i1 < p - 1; ++i1)
901  for (unsigned int i2 = 0; i2 < p - 1; ++i2)
902  for (unsigned int j1 = 0; j1 < q - 1; ++j1)
903  for (unsigned int j2 = 0; j2 < q - 1; ++j2)
904  if ((std::fabs(
905  this->unit_support_points[index_map_inverse[i1 + 1]][0] -
906  fe_q_other
907  ->unit_support_points[index_map_inverse_other[j1 + 1]]
908  [0]) < 1e-14) &&
909  (std::fabs(
910  this->unit_support_points[index_map_inverse[i2 + 1]][0] -
911  fe_q_other
912  ->unit_support_points[index_map_inverse_other[j2 + 1]]
913  [0]) < 1e-14))
914  identities.emplace_back(i1 * (p - 1) + i2, j1 * (q - 1) + j2);
915 
916  return identities;
917  }
918  else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
919  {
920  // the FE_Nothing has no degrees of freedom, so there are no
921  // equivalencies to be recorded
922  return std::vector<std::pair<unsigned int, unsigned int>>();
923  }
924  else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
925  {
926  // if the other element has no elements on faces at all,
927  // then it would be impossible to enforce any kind of
928  // continuity even if we knew exactly what kind of element
929  // we have -- simply because the other element declares
930  // that it is discontinuous because it has no DoFs on
931  // its faces. in that case, just state that we have no
932  // constraints to declare
933  return std::vector<std::pair<unsigned int, unsigned int>>();
934  }
935  else
936  {
937  Assert(false, ExcNotImplemented());
938  return std::vector<std::pair<unsigned int, unsigned int>>();
939  }
940 }
941 
942 
943 
944 //---------------------------------------------------------------------------
945 // Auxiliary functions
946 //---------------------------------------------------------------------------
947 
948 
949 
950 template <int dim, int spacedim>
951 void
953  const std::vector<Point<1>> &points)
954 {
955  const std::vector<unsigned int> &index_map_inverse =
957 
958  // We can compute the support points by computing the tensor
959  // product of the 1d set of points. We could do this by hand, but it's
960  // easier to just re-use functionality that's already been implemented
961  // for quadrature formulas.
962  const Quadrature<1> support_1d(points);
963  const Quadrature<dim> support_quadrature(support_1d); // NOLINT
964 
965  // The only thing we have to do is reorder the points from tensor
966  // product order to the order in which we enumerate DoFs on cells
967  this->unit_support_points.resize(support_quadrature.size());
968  for (unsigned int k = 0; k < support_quadrature.size(); ++k)
969  this->unit_support_points[index_map_inverse[k]] =
970  support_quadrature.point(k);
971 }
972 
973 
974 
975 template <int dim, int spacedim>
976 void
978  const std::vector<Point<1>> &points)
979 {
980  // TODO: the implementation makes the assumption that all faces have the
981  // same number of dofs
982  AssertDimension(this->n_unique_faces(), 1);
983  const unsigned int face_no = 0;
984 
985  this->unit_face_support_points[face_no].resize(
986  Utilities::fixed_power<dim - 1>(q_degree + 1));
987 
988  // In 1D, there is only one 0-dimensional support point, so there is nothing
989  // more to be done.
990  if (dim == 1)
991  return;
992 
993  // find renumbering of faces and assign from values of quadrature
994  const std::vector<unsigned int> face_index_map =
996 
997  // We can compute the support points by computing the tensor
998  // product of the 1d set of points. We could do this by hand, but it's
999  // easier to just re-use functionality that's already been implemented
1000  // for quadrature formulas.
1001  const Quadrature<1> support_1d(points);
1002  const Quadrature<dim - 1> support_quadrature(support_1d); // NOLINT
1003 
1004  // The only thing we have to do is reorder the points from tensor
1005  // product order to the order in which we enumerate DoFs on cells
1006  this->unit_face_support_points[face_no].resize(support_quadrature.size());
1007  for (unsigned int k = 0; k < support_quadrature.size(); ++k)
1008  this->unit_face_support_points[face_no][face_index_map[k]] =
1009  support_quadrature.point(k);
1010 }
1011 
1012 
1013 
1014 template <int dim, int spacedim>
1015 void
1017 {
1018  // for 1D and 2D, do nothing
1019  if (dim < 3)
1020  return;
1021 
1022  // TODO: the implementation makes the assumption that all faces have the
1023  // same number of dofs
1024  AssertDimension(this->n_unique_faces(), 1);
1025  const unsigned int face_no = 0;
1026 
1028  .n_elements() == 8 * this->n_dofs_per_quad(face_no),
1029  ExcInternalError());
1030 
1031  const unsigned int n = q_degree - 1;
1032  Assert(n * n == this->n_dofs_per_quad(face_no), ExcInternalError());
1033 
1034  // the dofs on a face are connected to a n x n matrix. for example, for
1035  // degree==4 we have the following dofs on a quad
1036 
1037  // ___________
1038  // | |
1039  // | 6 7 8 |
1040  // | |
1041  // | 3 4 5 |
1042  // | |
1043  // | 0 1 2 |
1044  // |___________|
1045  //
1046  // we have dof_no=i+n*j with index i in x-direction and index j in
1047  // y-direction running from 0 to n-1. to extract i and j we can use
1048  // i=dof_no%n and j=dof_no/n. i and j can then be used to construct the
1049  // rotated and mirrored numbers.
1050 
1051 
1052  for (unsigned int local = 0; local < this->n_dofs_per_quad(face_no); ++local)
1053  // face support points are in lexicographic ordering with x running
1054  // fastest. invert that (y running fastest)
1055  {
1056  unsigned int i = local % n, j = local / n;
1057 
1058  // face_orientation=false, face_flip=false, face_rotation=false
1060  0) =
1061  j + i * n - local;
1062  // face_orientation=false, face_flip=false, face_rotation=true
1064  1) =
1065  i + (n - 1 - j) * n - local;
1066  // face_orientation=false, face_flip=true, face_rotation=false
1068  2) =
1069  (n - 1 - j) + (n - 1 - i) * n - local;
1070  // face_orientation=false, face_flip=true, face_rotation=true
1072  3) =
1073  (n - 1 - i) + j * n - local;
1074  // face_orientation=true, face_flip=false, face_rotation=false
1076  4) = 0;
1077  // face_orientation=true, face_flip=false, face_rotation=true
1079  5) =
1080  j + (n - 1 - i) * n - local;
1081  // face_orientation=true, face_flip=true, face_rotation=false
1083  6) =
1084  (n - 1 - i) + (n - 1 - j) * n - local;
1085  // face_orientation=true, face_flip=true, face_rotation=true
1087  7) =
1088  (n - 1 - j) + i * n - local;
1089  }
1090 
1091  // additionally initialize reordering of line dofs
1092  for (unsigned int i = 0; i < this->n_dofs_per_line(); ++i)
1094  this->n_dofs_per_line() - 1 - i - i;
1095 }
1096 
1097 
1098 
1099 template <int dim, int spacedim>
1100 unsigned int
1101 FE_Q_Base<dim, spacedim>::face_to_cell_index(const unsigned int face_index,
1102  const unsigned int face,
1103  const bool face_orientation,
1104  const bool face_flip,
1105  const bool face_rotation) const
1106 {
1107  AssertIndexRange(face_index, this->n_dofs_per_face(face));
1109 
1110  // TODO: we could presumably solve the 3d case below using the
1111  // adjust_quad_dof_index_for_face_orientation_table field. for the
1112  // 2d case, we can't use adjust_line_dof_index_for_line_orientation_table
1113  // since that array is empty (presumably because we thought that
1114  // there are no flipped edges in 2d, but these can happen in
1115  // DoFTools::make_periodicity_constraints, for example). so we
1116  // would need to either fill this field, or rely on derived classes
1117  // implementing this function, as we currently do
1118 
1119  // we need to distinguish between DoFs on vertices, lines and in 3d quads.
1120  // do so in a sequence of if-else statements
1121  if (face_index < this->get_first_face_line_index(face))
1122  // DoF is on a vertex
1123  {
1124  // get the number of the vertex on the face that corresponds to this DoF,
1125  // along with the number of the DoF on this vertex
1126  const unsigned int face_vertex = face_index / this->n_dofs_per_vertex();
1127  const unsigned int dof_index_on_vertex =
1128  face_index % this->n_dofs_per_vertex();
1129 
1130  // then get the number of this vertex on the cell and translate
1131  // this to a DoF number on the cell
1133  face, face_vertex, face_orientation, face_flip, face_rotation) *
1134  this->n_dofs_per_vertex() +
1135  dof_index_on_vertex);
1136  }
1137  else if (face_index < this->get_first_face_quad_index(face))
1138  // DoF is on a face
1139  {
1140  // do the same kind of translation as before. we need to only consider
1141  // DoFs on the lines, i.e., ignoring those on the vertices
1142  const unsigned int index =
1143  face_index - this->get_first_face_line_index(face);
1144 
1145  const unsigned int face_line = index / this->n_dofs_per_line();
1146  const unsigned int dof_index_on_line = index % this->n_dofs_per_line();
1147 
1148  // we now also need to adjust the line index for the case of
1149  // face orientation, face flips, etc
1150  unsigned int adjusted_dof_index_on_line = 0;
1151  switch (dim)
1152  {
1153  case 1:
1154  Assert(false, ExcInternalError());
1155  break;
1156 
1157  case 2:
1158  // in 2d, only face_flip has a meaning. if it is set, consider
1159  // dofs in reverse order
1160  if (face_flip == false)
1161  adjusted_dof_index_on_line = dof_index_on_line;
1162  else
1163  adjusted_dof_index_on_line =
1164  this->n_dofs_per_line() - dof_index_on_line - 1;
1165  break;
1166 
1167  case 3:
1168  // in 3d, things are difficult. someone will have to think
1169  // about how this code here should look like, by drawing a bunch
1170  // of pictures of how all the faces can look like with the various
1171  // flips and rotations.
1172  //
1173  // that said, the Q2 case is easy enough to implement, as is the
1174  // case where everything is in standard orientation
1175  Assert((this->n_dofs_per_line() <= 1) ||
1176  ((face_orientation == true) && (face_flip == false) &&
1177  (face_rotation == false)),
1178  ExcNotImplemented());
1179  adjusted_dof_index_on_line = dof_index_on_line;
1180  break;
1181 
1182  default:
1183  Assert(false, ExcInternalError());
1184  }
1185 
1186  return (this->get_first_line_index() +
1188  face, face_line, face_orientation, face_flip, face_rotation) *
1189  this->n_dofs_per_line() +
1190  adjusted_dof_index_on_line);
1191  }
1192  else
1193  // DoF is on a quad
1194  {
1195  Assert(dim >= 3, ExcInternalError());
1196 
1197  // ignore vertex and line dofs
1198  const unsigned int index =
1199  face_index - this->get_first_face_quad_index(face);
1200 
1201  // the same is true here as above for the 3d case -- someone will
1202  // just have to draw a bunch of pictures. in the meantime,
1203  // we can implement the Q2 case in which it is simple
1204  Assert((this->n_dofs_per_quad(face) <= 1) ||
1205  ((face_orientation == true) && (face_flip == false) &&
1206  (face_rotation == false)),
1207  ExcNotImplemented());
1208  return (this->get_first_quad_index(face) + index);
1209  }
1210 }
1211 
1212 
1213 
1214 template <int dim, int spacedim>
1215 std::vector<unsigned int>
1217 {
1218  using FEQ = FE_Q_Base<dim, spacedim>;
1219  AssertThrow(degree > 0, typename FEQ::ExcFEQCannotHaveDegree0());
1220  std::vector<unsigned int> dpo(dim + 1, 1U);
1221  for (unsigned int i = 1; i < dpo.size(); ++i)
1222  dpo[i] = dpo[i - 1] * (degree - 1);
1223  return dpo;
1224 }
1225 
1226 
1227 
1228 template <int dim, int spacedim>
1229 void
1231  const std::vector<Point<1>> &points)
1232 {
1234 }
1235 
1236 
1237 
1238 template <int dim, int spacedim>
1239 const FullMatrix<double> &
1241  const unsigned int child,
1242  const RefinementCase<dim> &refinement_case) const
1243 {
1244  AssertIndexRange(refinement_case,
1246  Assert(refinement_case != RefinementCase<dim>::no_refinement,
1247  ExcMessage(
1248  "Prolongation matrices are only available for refined cells!"));
1249  AssertIndexRange(child, GeometryInfo<dim>::n_children(refinement_case));
1250 
1251  // initialization upon first request
1252  if (this->prolongation[refinement_case - 1][child].n() == 0)
1253  {
1254  std::lock_guard<std::mutex> lock(this->mutex);
1255 
1256  // if matrix got updated while waiting for the lock
1257  if (this->prolongation[refinement_case - 1][child].n() ==
1258  this->n_dofs_per_cell())
1259  return this->prolongation[refinement_case - 1][child];
1260 
1261  // distinguish q/q_dg0 case: only treat Q dofs first
1262  const unsigned int q_dofs_per_cell =
1263  Utilities::fixed_power<dim>(q_degree + 1);
1264 
1265  // compute the interpolation matrices in much the same way as we do for
1266  // the constraints. it's actually simpler here, since we don't have this
1267  // weird renumbering stuff going on. The trick is again that we the
1268  // interpolation matrix is formed by a permutation of the indices of the
1269  // cell matrix. The value eps is used a threshold to decide when certain
1270  // evaluations of the Lagrange polynomials are zero or one.
1271  const double eps = 1e-15 * q_degree * dim;
1272 
1273 #ifdef DEBUG
1274  // in DEBUG mode, check that the evaluation of support points in the
1275  // current numbering gives the identity operation
1276  for (unsigned int i = 0; i < q_dofs_per_cell; ++i)
1277  {
1278  Assert(std::fabs(1. - this->poly_space->compute_value(
1279  i, this->unit_support_points[i])) < eps,
1280  ExcInternalError("The Lagrange polynomial does not evaluate "
1281  "to one or zero in a nodal point. "
1282  "This typically indicates that the "
1283  "polynomial interpolation is "
1284  "ill-conditioned such that round-off "
1285  "prevents the sum to be one."));
1286  for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
1287  if (j != i)
1288  Assert(std::fabs(this->poly_space->compute_value(
1289  i, this->unit_support_points[j])) < eps,
1291  "The Lagrange polynomial does not evaluate "
1292  "to one or zero in a nodal point. "
1293  "This typically indicates that the "
1294  "polynomial interpolation is "
1295  "ill-conditioned such that round-off "
1296  "prevents the sum to be one."));
1297  }
1298 #endif
1299 
1300  // to efficiently evaluate the polynomial at the subcell, make use of
1301  // the tensor product structure of this element and only evaluate 1D
1302  // information from the polynomial. This makes the cost of this function
1303  // almost negligible also for high order elements
1304  const unsigned int dofs1d = q_degree + 1;
1305  std::vector<Table<2, double>> subcell_evaluations(
1306  dim, Table<2, double>(dofs1d, dofs1d));
1307 
1308  const std::vector<unsigned int> &index_map_inverse =
1310 
1311  // helper value: step size how to walk through diagonal and how many
1312  // points we have left apart from the first dimension
1313  unsigned int step_size_diag = 0;
1314  {
1315  unsigned int factor = 1;
1316  for (unsigned int d = 0; d < dim; ++d)
1317  {
1318  step_size_diag += factor;
1319  factor *= dofs1d;
1320  }
1321  }
1322 
1323  FullMatrix<double> prolongate(this->n_dofs_per_cell(),
1324  this->n_dofs_per_cell());
1325 
1326  // go through the points in diagonal to capture variation in all
1327  // directions simultaneously
1328  for (unsigned int j = 0; j < dofs1d; ++j)
1329  {
1330  const unsigned int diag_comp = index_map_inverse[j * step_size_diag];
1331  const Point<dim> p_subcell = this->unit_support_points[diag_comp];
1332  const Point<dim> p_cell =
1334  child,
1335  refinement_case);
1336  for (unsigned int i = 0; i < dofs1d; ++i)
1337  for (unsigned int d = 0; d < dim; ++d)
1338  {
1339  // evaluate along line where only x is different from zero
1340  Point<dim> point;
1341  point[0] = p_cell[d];
1342  const double cell_value =
1343  this->poly_space->compute_value(index_map_inverse[i], point);
1344 
1345  // cut off values that are too small. note that we have here
1346  // Lagrange interpolation functions, so they should be zero at
1347  // almost all points, and one at the others, at least on the
1348  // subcells. so set them to their exact values
1349  //
1350  // the actual cut-off value is somewhat fuzzy, but it works
1351  // for 2e-13*degree*dim (see above), which is kind of
1352  // reasonable given that we compute the values of the
1353  // polynomials via an degree-step recursion and then multiply
1354  // the 1d-values. this gives us a linear growth in degree*dim,
1355  // times a small constant.
1356  //
1357  // the embedding matrix is given by applying the inverse of
1358  // the subcell matrix on the cell_interpolation matrix. since
1359  // the subcell matrix is actually only a permutation vector,
1360  // all we need to do is to switch the rows we write the data
1361  // into. moreover, cut off very small values here
1362  if (std::fabs(cell_value) < eps)
1363  subcell_evaluations[d](j, i) = 0;
1364  else
1365  subcell_evaluations[d](j, i) = cell_value;
1366  }
1367  }
1368 
1369  // now expand from 1D info. block innermost dimension (x_0) in order to
1370  // avoid difficult checks at innermost loop
1371  unsigned int j_indices[dim];
1372  internal::FE_Q_Base::zero_indices<dim>(j_indices);
1373  for (unsigned int j = 0; j < q_dofs_per_cell; j += dofs1d)
1374  {
1375  unsigned int i_indices[dim];
1376  internal::FE_Q_Base::zero_indices<dim>(i_indices);
1377  for (unsigned int i = 0; i < q_dofs_per_cell; i += dofs1d)
1378  {
1379  double val_extra_dim = 1.;
1380  for (unsigned int d = 1; d < dim; ++d)
1381  val_extra_dim *=
1382  subcell_evaluations[d](j_indices[d - 1], i_indices[d - 1]);
1383 
1384  // innermost sum where we actually compute. the same as
1385  // prolongate(j,i) = this->poly_space->compute_value (i, p_cell)
1386  for (unsigned int jj = 0; jj < dofs1d; ++jj)
1387  {
1388  const unsigned int j_ind = index_map_inverse[j + jj];
1389  for (unsigned int ii = 0; ii < dofs1d; ++ii)
1390  prolongate(j_ind, index_map_inverse[i + ii]) =
1391  val_extra_dim * subcell_evaluations[0](jj, ii);
1392  }
1393 
1394  // update indices that denote the tensor product position. a bit
1395  // fuzzy and therefore not done for innermost x_0 direction
1396  internal::FE_Q_Base::increment_indices<dim>(i_indices, dofs1d);
1397  }
1398  Assert(i_indices[dim - 1] == 1, ExcInternalError());
1399  internal::FE_Q_Base::increment_indices<dim>(j_indices, dofs1d);
1400  }
1401 
1402  // the discontinuous node is simply mapped on the discontinuous node on
1403  // the child element
1404  if (q_dofs_per_cell < this->n_dofs_per_cell())
1405  prolongate(q_dofs_per_cell, q_dofs_per_cell) = 1.;
1406 
1407  // and make sure that the row sum is 1. this must be so since for this
1408  // element, the shape functions add up to one
1409 #ifdef DEBUG
1410  for (unsigned int row = 0; row < this->n_dofs_per_cell(); ++row)
1411  {
1412  double sum = 0;
1413  for (unsigned int col = 0; col < this->n_dofs_per_cell(); ++col)
1414  sum += prolongate(row, col);
1415  Assert(std::fabs(sum - 1.) <
1416  std::max(eps, 5e-16 * std::sqrt(this->n_dofs_per_cell())),
1417  ExcInternalError("The entries in a row of the local "
1418  "prolongation matrix do not add to one. "
1419  "This typically indicates that the "
1420  "polynomial interpolation is "
1421  "ill-conditioned such that round-off "
1422  "prevents the sum to be one."));
1423  }
1424 #endif
1425 
1426  // swap matrices
1427  prolongate.swap(const_cast<FullMatrix<double> &>(
1428  this->prolongation[refinement_case - 1][child]));
1429  }
1430 
1431  // finally return the matrix
1432  return this->prolongation[refinement_case - 1][child];
1433 }
1434 
1435 
1436 
1437 template <int dim, int spacedim>
1438 const FullMatrix<double> &
1440  const unsigned int child,
1441  const RefinementCase<dim> &refinement_case) const
1442 {
1443  AssertIndexRange(refinement_case,
1445  Assert(refinement_case != RefinementCase<dim>::no_refinement,
1446  ExcMessage(
1447  "Restriction matrices are only available for refined cells!"));
1448  AssertIndexRange(child, GeometryInfo<dim>::n_children(refinement_case));
1449 
1450  // initialization upon first request
1451  if (this->restriction[refinement_case - 1][child].n() == 0)
1452  {
1453  std::lock_guard<std::mutex> lock(this->mutex);
1454 
1455  // if matrix got updated while waiting for the lock...
1456  if (this->restriction[refinement_case - 1][child].n() ==
1457  this->n_dofs_per_cell())
1458  return this->restriction[refinement_case - 1][child];
1459 
1460  FullMatrix<double> my_restriction(this->n_dofs_per_cell(),
1461  this->n_dofs_per_cell());
1462  // distinguish q/q_dg0 case
1463  const unsigned int q_dofs_per_cell =
1464  Utilities::fixed_power<dim>(q_degree + 1);
1465 
1466  // for Lagrange interpolation polynomials based on equidistant points,
1467  // construction of the restriction matrices is relatively simple. the
1468  // reason is that in this case the interpolation points on the mother
1469  // cell are always also interpolation points for some shape function on
1470  // one or the other child.
1471  //
1472  // in the general case with non-equidistant points, we need to actually
1473  // do an interpolation. thus, we take the interpolation points on the
1474  // mother cell and evaluate the shape functions of the child cell on
1475  // those points. it does not hurt in the equidistant case because then
1476  // simple one shape function evaluates to one and the others to zero.
1477  //
1478  // this element is non-additive in all its degrees of freedom by
1479  // default, which requires care in downstream use. fortunately, even the
1480  // interpolation on non-equidistant points is invariant under the
1481  // assumption that whenever a row makes a non-zero contribution to the
1482  // mother's residual, the correct value is interpolated.
1483 
1484  const double eps = 1e-15 * q_degree * dim;
1485  const std::vector<unsigned int> &index_map_inverse =
1487 
1488  const unsigned int dofs1d = q_degree + 1;
1489  std::vector<Tensor<1, dim>> evaluations1d(dofs1d);
1490 
1491  my_restriction.reinit(this->n_dofs_per_cell(), this->n_dofs_per_cell());
1492 
1493  for (unsigned int i = 0; i < q_dofs_per_cell; ++i)
1494  {
1495  unsigned int mother_dof = index_map_inverse[i];
1496  const Point<dim> p_cell = this->unit_support_points[mother_dof];
1497 
1498  // check whether this interpolation point is inside this child cell
1499  const Point<dim> p_subcell =
1501  child,
1502  refinement_case);
1504  {
1505  // same logic as in initialize_embedding to evaluate the
1506  // polynomial faster than from the tensor product: since we
1507  // evaluate all polynomials, it is much faster to just compute
1508  // the 1D values for all polynomials before and then get the
1509  // dim-data.
1510  for (unsigned int j = 0; j < dofs1d; ++j)
1511  for (unsigned int d = 0; d < dim; ++d)
1512  {
1513  Point<dim> point;
1514  point[0] = p_subcell[d];
1515  evaluations1d[j][d] =
1516  this->poly_space->compute_value(index_map_inverse[j],
1517  point);
1518  }
1519  unsigned int j_indices[dim];
1520  internal::FE_Q_Base::zero_indices<dim>(j_indices);
1521  double sum_check = 0;
1522  for (unsigned int j = 0; j < q_dofs_per_cell; j += dofs1d)
1523  {
1524  double val_extra_dim = 1.;
1525  for (unsigned int d = 1; d < dim; ++d)
1526  val_extra_dim *= evaluations1d[j_indices[d - 1]][d];
1527  for (unsigned int jj = 0; jj < dofs1d; ++jj)
1528  {
1529  // find the child shape function(s) corresponding to
1530  // this point. Usually this is just one function;
1531  // however, when we use FE_Q on arbitrary nodes a parent
1532  // support point might not be a child support point, and
1533  // then we will get more than one nonzero value per
1534  // row. Still, the values should sum up to 1
1535  const double val = val_extra_dim * evaluations1d[jj][0];
1536  const unsigned int child_dof = index_map_inverse[j + jj];
1537  if (std::fabs(val - 1.) < eps)
1538  my_restriction(mother_dof, child_dof) = 1.;
1539  else if (std::fabs(val) > eps)
1540  my_restriction(mother_dof, child_dof) = val;
1541  sum_check += val;
1542  }
1543  internal::FE_Q_Base::increment_indices<dim>(j_indices,
1544  dofs1d);
1545  }
1546  Assert(std::fabs(sum_check - 1) <
1547  std::max(eps,
1548  5e-16 * std::sqrt(this->n_dofs_per_cell())),
1549  ExcInternalError("The entries in a row of the local "
1550  "restriction matrix do not add to one. "
1551  "This typically indicates that the "
1552  "polynomial interpolation is "
1553  "ill-conditioned such that round-off "
1554  "prevents the sum to be one."));
1555  }
1556 
1557  // part for FE_Q_DG0
1558  if (q_dofs_per_cell < this->n_dofs_per_cell())
1559  my_restriction(this->n_dofs_per_cell() - 1,
1560  this->n_dofs_per_cell() - 1) =
1562  RefinementCase<dim>(refinement_case));
1563  }
1564 
1565  // swap the just computed restriction matrix into the
1566  // element of the vector stored in the base class
1567  my_restriction.swap(const_cast<FullMatrix<double> &>(
1568  this->restriction[refinement_case - 1][child]));
1569  }
1570 
1571  return this->restriction[refinement_case - 1][child];
1572 }
1573 
1574 
1575 
1576 //---------------------------------------------------------------------------
1577 // Data field initialization
1578 //---------------------------------------------------------------------------
1579 
1580 
1581 template <int dim, int spacedim>
1582 bool
1584  const unsigned int shape_index,
1585  const unsigned int face_index) const
1586 {
1587  AssertIndexRange(shape_index, this->n_dofs_per_cell());
1589 
1590  // in 1d, things are simple. since there is only one degree of freedom per
1591  // vertex in this class, the first is on vertex 0 (==face 0 in some sense),
1592  // the second on face 1:
1593  if (dim == 1)
1594  return (((shape_index == 0) && (face_index == 0)) ||
1595  ((shape_index == 1) && (face_index == 1)));
1596 
1597  // first, special-case interior shape functions, since they have no support
1598  // no-where on the boundary
1599  if (((dim == 2) &&
1600  (shape_index >= this->get_first_quad_index(0 /*first quad*/))) ||
1601  ((dim == 3) && (shape_index >= this->get_first_hex_index())))
1602  return false;
1603 
1604  // let's see whether this is a vertex
1605  if (shape_index < this->get_first_line_index())
1606  {
1607  // for Q elements, there is one dof per vertex, so
1608  // shape_index==vertex_number. check whether this vertex is on the given
1609  // face. thus, for each face, give a list of vertices
1610  const unsigned int vertex_no = shape_index;
1612  ExcInternalError());
1613 
1614  for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_face; ++v)
1615  if (GeometryInfo<dim>::face_to_cell_vertices(face_index, v) ==
1616  vertex_no)
1617  return true;
1618 
1619  return false;
1620  }
1621  else if (shape_index < this->get_first_quad_index(0 /*first quad*/))
1622  // ok, dof is on a line
1623  {
1624  const unsigned int line_index =
1625  (shape_index - this->get_first_line_index()) / this->n_dofs_per_line();
1627  ExcInternalError());
1628 
1629  // in 2d, the line is the face, so get the line index
1630  if (dim == 2)
1631  return (line_index == face_index);
1632  else if (dim == 3)
1633  {
1634  // silence compiler warning
1635  const unsigned int lines_per_face =
1636  dim == 3 ? GeometryInfo<dim>::lines_per_face : 1;
1637  // see whether the given line is on the given face.
1638  for (unsigned int l = 0; l < lines_per_face; ++l)
1639  if (GeometryInfo<3>::face_to_cell_lines(face_index, l) ==
1640  line_index)
1641  return true;
1642 
1643  return false;
1644  }
1645  else
1646  Assert(false, ExcNotImplemented());
1647  }
1648  else if (shape_index < this->get_first_hex_index())
1649  // dof is on a quad
1650  {
1651  const unsigned int quad_index =
1652  (shape_index - this->get_first_quad_index(0)) /
1653  this->n_dofs_per_quad(face_index); // this won't work
1654  Assert(static_cast<signed int>(quad_index) <
1655  static_cast<signed int>(GeometryInfo<dim>::quads_per_cell),
1656  ExcInternalError());
1657 
1658  // in 2d, cell bubble are zero on all faces. but we have treated this
1659  // case above already
1660  Assert(dim != 2, ExcInternalError());
1661 
1662  // in 3d, quad_index=face_index
1663  if (dim == 3)
1664  return (quad_index == face_index);
1665  else
1666  Assert(false, ExcNotImplemented());
1667  }
1668  else
1669  // dof on hex
1670  {
1671  // can only happen in 3d, but this case has already been covered above
1672  Assert(false, ExcNotImplemented());
1673  return false;
1674  }
1675 
1676  // we should not have gotten here
1677  Assert(false, ExcInternalError());
1678  return false;
1679 }
1680 
1681 
1682 
1683 template <int dim, int spacedim>
1684 std::pair<Table<2, bool>, std::vector<unsigned int>>
1686 {
1687  Table<2, bool> constant_modes(1, this->n_dofs_per_cell());
1688  // We here just care for the constant mode due to the polynomial space
1689  // without any enrichments
1690  // As there may be more constant modes derived classes may to implement this
1691  // themselves. An example for this is FE_Q_DG0.
1692  for (unsigned int i = 0; i < Utilities::fixed_power<dim>(q_degree + 1); ++i)
1693  constant_modes(0, i) = true;
1694  return std::pair<Table<2, bool>, std::vector<unsigned int>>(
1695  constant_modes, std::vector<unsigned int>(1, 0));
1696 }
1697 
1698 
1699 
1700 // explicit instantiations
1701 #include "fe_q_base.inst"
1702 
static Point< dim > child_to_cell_coordinates(const Point< dim > &p, const unsigned int child_index, const RefinementCase< dim > refine_case=RefinementCase< dim >::isotropic_refinement)
size_type m() const
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_vertex_dof_identities(const FiniteElement< dim, spacedim > &fe_other) const override
Definition: fe_q_base.cc:737
static const unsigned int invalid_unsigned_int
Definition: types.h:196
std::vector< std::vector< FullMatrix< double > > > restriction
Definition: fe.h:2404
void initialize_unit_face_support_points(const std::vector< Point< 1 >> &points)
Definition: fe_q_base.cc:977
#define AssertDimension(dim1, dim2)
Definition: exceptions.h:1657
void initialize_unit_support_points(const std::vector< Point< 1 >> &points)
Definition: fe_q_base.cc:952
unsigned int get_first_line_index() const
FullMatrix< double > interface_constraints
Definition: fe.h:2430
static void project_to_subface(const SubQuadrature &quadrature, const unsigned int face_no, const unsigned int subface_no, std::vector< Point< dim >> &q_points, const RefinementCase< dim - 1 > &ref_case=RefinementCase< dim - 1 >::isotropic_refinement)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
Task< RT > new_task(const std::function< RT()> &function)
virtual bool hp_constraints_are_implemented() const override
Definition: fe_q_base.cc:728
std::vector< unsigned int > lexicographic_to_hierarchic_numbering(unsigned int degree)
bool has_face_support_points(const unsigned int face_no=0) const
Definition: fe.cc:1127
#define AssertIndexRange(index, range)
Definition: exceptions.h:1722
const unsigned int degree
Definition: fe_base.h:435
const Point< dim > & point(const unsigned int i) const
unsigned int n_dofs_per_vertex() const
void set_numbering(const std::vector< unsigned int > &renumber)
static Point< dim > cell_to_child_coordinates(const Point< dim > &p, const unsigned int child_index, const RefinementCase< dim > refine_case=RefinementCase< dim >::isotropic_refinement)
STL namespace.
static const char U
void initialize_quad_dof_index_permutation()
Definition: fe_q_base.cc:1016
virtual const FullMatrix< double > & get_restriction_matrix(const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
Definition: fe_q_base.cc:1439
unsigned int get_first_face_line_index(const unsigned int face_no=0) const
#define AssertThrow(cond, exc)
Definition: exceptions.h:1575
unsigned int n_dofs_per_quad(unsigned int face_no=0) const
static ::ExceptionBase & ExcInterpolationNotImplemented()
unsigned int get_first_face_quad_index(const unsigned int face_no=0) const
void set_numbering(const std::vector< unsigned int > &renumber)
const ScalarPolynomialsBase< dim > & get_poly_space() const
void initialize(const std::vector< Point< 1 >> &support_points_1d)
Definition: fe_q_base.cc:435
std::vector< Point< dim > > unit_support_points
Definition: fe.h:2442
virtual void get_subface_interpolation_matrix(const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix, const unsigned int face_no=0) const override
Definition: fe_q_base.cc:624
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_line_dof_identities(const FiniteElement< dim, spacedim > &fe_other) const override
Definition: fe_q_base.cc:781
FE_Q_Base(const ScalarPolynomialsBase< dim > &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags)
Definition: fe_q_base.cc:416
virtual double shape_value(const unsigned int i, const Point< dim > &p) const override
static ::ExceptionBase & ExcMessage(std::string arg1)
size_type n() const
unsigned int n_dofs_per_line() const
std::vector< std::vector< FullMatrix< double > > > prolongation
Definition: fe.h:2418
void initialize_constraints(const std::vector< Point< 1 >> &points)
Definition: fe_q_base.cc:1230
T sum(const T &t, const MPI_Comm &mpi_communicator)
void set_numbering(const std::vector< unsigned int > &renumber)
virtual unsigned int face_to_cell_index(const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const override
Definition: fe_q_base.cc:1101
unsigned int get_first_hex_index() const
#define Assert(cond, exc)
Definition: exceptions.h:1465
static ::ExceptionBase & ExcDimensionMismatch(std::size_t arg1, std::size_t arg2)
std::vector< unsigned int > get_poly_space_numbering_inverse() const
const std::vector< Point< dim - 1 > > & get_unit_face_support_points(const unsigned int face_no=0) const
Definition: fe.cc:1108
virtual void get_interpolation_matrix(const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const override
Definition: fe_q_base.cc:511
std::vector< std::vector< Point< dim - 1 > > > unit_face_support_points
Definition: fe.h:2449
#define DEAL_II_NAMESPACE_CLOSE
Definition: config.h:399
virtual bool has_support_on_face(const unsigned int shape_index, const unsigned int face_index) const override
Definition: fe_q_base.cc:1583
Expression fabs(const Expression &x)
unsigned int get_first_quad_index(const unsigned int quad_no=0) const
unsigned int n_dofs_per_face(unsigned int face_no=0, unsigned int child=0) const
static void initialize_constraints(const std::vector< Point< 1 >> &, FE_Q_Base< 2, spacedim > &fe)
Definition: fe_q_base.cc:105
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition: utilities.cc:188
unsigned int n_unique_faces() const
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
unsigned int size() const
static void initialize_constraints(const std::vector< Point< 1 >> &, FE_Q_Base< 3, spacedim > &fe)
Definition: fe_q_base.cc:216
static unsigned int n_children(const RefinementCase< dim > &refinement_case)
static void project_to_face(const SubQuadrature &quadrature, const unsigned int face_no, std::vector< Point< dim >> &q_points)
virtual void get_face_interpolation_matrix(const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix, const unsigned int face_no=0) const override
Definition: fe_q_base.cc:609
unsigned int n_components() const
unsigned int n_dofs_per_cell() const
#define DEAL_II_NAMESPACE_OPEN
Definition: config.h:398
Threads::Mutex mutex
Definition: fe_q_base.h:337
const unsigned int q_degree
Definition: fe_q_base.h:344
static ::ExceptionBase & ExcNotImplemented()
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > get_constant_modes() const override
Definition: fe_q_base.cc:1685
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_quad_dof_identities(const FiniteElement< dim, spacedim > &fe_other, const unsigned int face_no=0) const override
Definition: fe_q_base.cc:876
const std::unique_ptr< ScalarPolynomialsBase< dim > > poly_space
Definition: fe_poly.h:534
static std::vector< unsigned int > get_dpo_vector(const unsigned int degree)
Definition: fe_q_base.cc:1216
static void initialize_constraints(const std::vector< Point< 1 >> &, FE_Q_Base< 1, spacedim > &)
Definition: fe_q_base.cc:96
std::vector< int > adjust_line_dof_index_for_line_orientation_table
Definition: fe.h:2493
TableIndices< 2 > interface_constraints_size() const
Definition: fe.cc:899
virtual const FullMatrix< double > & get_prolongation_matrix(const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
Definition: fe_q_base.cc:1240
ReferenceCell reference_cell() const
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
std::vector< Table< 2, int > > adjust_quad_dof_index_for_face_orientation_table
Definition: fe.h:2478
static ::ExceptionBase & ExcInternalError()