Reference documentation for deal.II version GIT relicensing-897-g3473bfcca7 2024-06-24 11:40:02+00:00
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
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fe_q_base.cc
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1// ------------------------------------------------------------------------
2//
3// SPDX-License-Identifier: LGPL-2.1-or-later
4// Copyright (C) 2013 - 2024 by the deal.II authors
5//
6// This file is part of the deal.II library.
7//
8// Part of the source code is dual licensed under Apache-2.0 WITH
9// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
10// governing the source code and code contributions can be found in
11// LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
12//
13// ------------------------------------------------------------------------
14
15
25
26#include <deal.II/fe/fe_dgp.h>
27#include <deal.II/fe/fe_dgq.h>
34#include <deal.II/fe/fe_tools.h>
36
37#include <memory>
38#include <sstream>
39#include <vector>
40
42
43
44namespace internal
45{
46 namespace FE_Q_Base
47 {
48 namespace
49 {
50 // in get_restriction_matrix() and get_prolongation_matrix(), want to undo
51 // tensorization on inner loops for performance reasons. this clears a
52 // dim-array
53 template <int dim>
54 inline void
55 zero_indices(unsigned int (&indices)[dim])
56 {
57 for (unsigned int d = 0; d < dim; ++d)
58 indices[d] = 0;
59 }
60
61
62
63 // in get_restriction_matrix() and get_prolongation_matrix(), want to undo
64 // tensorization on inner loops for performance reasons. this increments
65 // tensor product indices
66 template <int dim>
67 inline void
68 increment_indices(unsigned int (&indices)[dim], const unsigned int dofs1d)
69 {
70 ++indices[0];
71 for (unsigned int d = 0; d < dim - 1; ++d)
72 if (indices[d] == dofs1d)
73 {
74 indices[d] = 0;
75 indices[d + 1]++;
76 }
77 }
78 } // namespace
79 } // namespace FE_Q_Base
80} // namespace internal
81
82
83
88template <int xdim, int xspacedim>
89struct FE_Q_Base<xdim, xspacedim>::Implementation
90{
95 template <int spacedim>
96 static void
97 initialize_constraints(const std::vector<Point<1>> &,
99 {
100 // no constraints in 1d
101 }
102
103
104 template <int spacedim>
105 static void
106 initialize_constraints(const std::vector<Point<1>> & /*points*/,
108 {
109 const unsigned int dim = 2;
110
111 unsigned int q_deg = fe.degree;
112 if (dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
113 &fe.get_poly_space()) != nullptr)
114 q_deg = fe.degree - 1;
115
116 // restricted to each face, the traces of the shape functions is an
117 // element of P_{k} (in 2d), or Q_{k} (in 3d), where k is the degree of
118 // the element. from this, we interpolate between mother and cell face.
119
120 // the interpolation process works as follows: on each subface, we want
121 // that finite element solutions from both sides coincide. i.e. if a and b
122 // are expansion coefficients for the shape functions from both sides, we
123 // seek a relation between a and b such that
124 // sum_j a_j phi^c_j(x) == sum_j b_j phi_j(x)
125 // for all points x on the interface. here, phi^c_j are the shape
126 // functions on the small cell on one side of the face, and phi_j those on
127 // the big cell on the other side. To get this relation, it suffices to
128 // look at a sufficient number of points for which this has to hold. if
129 // there are n functions, then we need n evaluation points, and we choose
130 // them equidistantly.
131 //
132 // we obtain the matrix system
133 // A a == B b
134 // where
135 // A_ij = phi^c_j(x_i)
136 // B_ij = phi_j(x_i)
137 // and the relation we are looking for is
138 // a = A^-1 B b
139 //
140 // for the special case of Lagrange interpolation polynomials, A_ij
141 // reduces to delta_ij, and
142 // a_i = B_ij b_j
143 // Hence, interface_constraints(i,j)=B_ij.
144 //
145 // for the general case, where we don't have Lagrange interpolation
146 // polynomials, this is a little more complicated. Then we would evaluate
147 // at a number of points and invert the interpolation matrix A.
148 //
149 // Note, that we build up these matrices for all subfaces at once, rather
150 // than considering them separately. the reason is that we finally will
151 // want to have them in this order anyway, as this is the format we need
152 // inside deal.II
153
154 // In the following the points x_i are constructed in following order
155 // (n=degree-1)
156 // *----------*---------*
157 // 1..n 0 n+1..2n
158 // i.e. first the midpoint of the line, then the support points on subface
159 // 0 and on subface 1
160 std::vector<Point<dim - 1>> constraint_points;
161 // Add midpoint
162 constraint_points.emplace_back(0.5);
163
164 if (q_deg > 1)
165 {
166 const unsigned int n = q_deg - 1;
167 const double step = 1. / q_deg;
168 // subface 0
169 for (unsigned int i = 1; i <= n; ++i)
170 constraint_points.push_back(
172 Point<dim - 1>(i * step), 0));
173 // subface 1
174 for (unsigned int i = 1; i <= n; ++i)
175 constraint_points.push_back(
177 Point<dim - 1>(i * step), 1));
178 }
179
180 // Now construct relation between destination (child) and source (mother)
181 // dofs.
182
183 fe.interface_constraints.TableBase<2, double>::reinit(
185
186 // use that the element evaluates to 1 at index 0 and along the line at
187 // zero
188 const std::vector<unsigned int> &index_map_inverse =
190 const std::vector<unsigned int> face_index_map =
193 fe.poly_space->compute_value(index_map_inverse[0], Point<dim>()) -
194 1.) < 1e-14,
196
197 for (unsigned int i = 0; i < constraint_points.size(); ++i)
198 for (unsigned int j = 0; j < q_deg + 1; ++j)
199 {
200 Point<dim> p;
201 p[0] = constraint_points[i][0];
202 fe.interface_constraints(i, face_index_map[j]) =
203 fe.poly_space->compute_value(index_map_inverse[j], p);
204
205 // if the value is small up to round-off, then simply set it to zero
206 // to avoid unwanted fill-in of the constraint matrices (which would
207 // then increase the number of other DoFs a constrained DoF would
208 // couple to)
209 if (std::fabs(fe.interface_constraints(i, face_index_map[j])) < 1e-13)
210 fe.interface_constraints(i, face_index_map[j]) = 0;
211 }
212 }
213
214
215 template <int spacedim>
216 static void
217 initialize_constraints(const std::vector<Point<1>> & /*points*/,
219 {
220 const unsigned int dim = 3;
221
222 unsigned int q_deg = fe.degree;
223 if (dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
224 &fe.get_poly_space()) != nullptr)
225 q_deg = fe.degree - 1;
226
227 // For a detailed documentation of the interpolation see the
228 // FE_Q_Base<2>::initialize_constraints function.
229
230 // In the following the points x_i are constructed in the order as
231 // described in the documentation of the FiniteElement class (fe_base.h),
232 // i.e.
233 // *--15--4--16--*
234 // | | |
235 // 10 19 6 20 12
236 // | | |
237 // 1--7---0--8---2
238 // | | |
239 // 9 17 5 18 11
240 // | | |
241 // *--13--3--14--*
242 std::vector<Point<dim - 1>> constraint_points;
243
244 // Add midpoint
245 constraint_points.emplace_back(0.5, 0.5);
246
247 // Add midpoints of lines of "mother-face"
248 constraint_points.emplace_back(0, 0.5);
249 constraint_points.emplace_back(1, 0.5);
250 constraint_points.emplace_back(0.5, 0);
251 constraint_points.emplace_back(0.5, 1);
252
253 if (q_deg > 1)
254 {
255 const unsigned int n = q_deg - 1;
256 const double step = 1. / q_deg;
257 std::vector<Point<dim - 2>> line_support_points(n);
258 for (unsigned int i = 0; i < n; ++i)
259 line_support_points[i][0] = (i + 1) * step;
260 const Quadrature<dim - 2> qline(line_support_points);
261
262 // auxiliary points in 2d
263 std::vector<Point<dim - 1>> p_line(n);
264
265 // Add nodes of lines interior in the "mother-face"
266
267 // line 5: use line 9
269 ReferenceCells::get_hypercube<dim - 1>(), qline, 0, 0, p_line);
270 for (unsigned int i = 0; i < n; ++i)
271 constraint_points.push_back(p_line[i] + Point<dim - 1>(0.5, 0));
272 // line 6: use line 10
274 ReferenceCells::get_hypercube<dim - 1>(), qline, 0, 1, p_line);
275 for (unsigned int i = 0; i < n; ++i)
276 constraint_points.push_back(p_line[i] + Point<dim - 1>(0.5, 0));
277 // line 7: use line 13
279 ReferenceCells::get_hypercube<dim - 1>(), qline, 2, 0, p_line);
280 for (unsigned int i = 0; i < n; ++i)
281 constraint_points.push_back(p_line[i] + Point<dim - 1>(0, 0.5));
282 // line 8: use line 14
284 ReferenceCells::get_hypercube<dim - 1>(), qline, 2, 1, p_line);
285 for (unsigned int i = 0; i < n; ++i)
286 constraint_points.push_back(p_line[i] + Point<dim - 1>(0, 0.5));
287
288 // DoFs on bordering lines lines 9-16
289 for (unsigned int face = 0;
290 face < GeometryInfo<dim - 1>::faces_per_cell;
291 ++face)
292 for (unsigned int subface = 0;
293 subface < GeometryInfo<dim - 1>::max_children_per_face;
294 ++subface)
295 {
297 ReferenceCells::get_hypercube<dim - 1>(),
298 qline,
299 face,
300 subface,
301 p_line);
302 constraint_points.insert(constraint_points.end(),
303 p_line.begin(),
304 p_line.end());
305 }
306
307 // Create constraints for interior nodes
308 std::vector<Point<dim - 1>> inner_points(n * n);
309 for (unsigned int i = 0, iy = 1; iy <= n; ++iy)
310 for (unsigned int ix = 1; ix <= n; ++ix)
311 inner_points[i++] = Point<dim - 1>(ix * step, iy * step);
312
313 // at the moment do this for isotropic face refinement only
314 for (unsigned int child = 0;
315 child < GeometryInfo<dim - 1>::max_children_per_cell;
316 ++child)
317 for (const auto &inner_point : inner_points)
318 constraint_points.push_back(
320 child));
321 }
322
323 // Now construct relation between destination (child) and source (mother)
324 // dofs.
325 const unsigned int pnts = (q_deg + 1) * (q_deg + 1);
326
327 // use that the element evaluates to 1 at index 0 and along the line at
328 // zero
329 const std::vector<unsigned int> &index_map_inverse =
331 const std::vector<unsigned int> face_index_map =
334 fe.poly_space->compute_value(index_map_inverse[0], Point<dim>()) -
335 1.) < 1e-14,
337
338 fe.interface_constraints.TableBase<2, double>::reinit(
340
341 for (unsigned int i = 0; i < constraint_points.size(); ++i)
342 {
343 const double interval = static_cast<double>(q_deg * 2);
344 bool mirror[dim - 1];
345 Point<dim> constraint_point;
346
347 // Eliminate FP errors in constraint points. Due to their origin, they
348 // must all be fractions of the unit interval. If we have polynomial
349 // degree 4, the refined element has 8 intervals. Hence the
350 // coordinates must be 0, 0.125, 0.25, 0.375 etc. Now the coordinates
351 // of the constraint points will be multiplied by the inverse of the
352 // interval size (in the example by 8). After that the coordinates
353 // must be integral numbers. Hence a normal truncation is performed
354 // and the coordinates will be scaled back. The equal treatment of all
355 // coordinates should eliminate any FP errors.
356 for (unsigned int k = 0; k < dim - 1; ++k)
357 {
358 const int coord_int =
359 static_cast<int>(constraint_points[i][k] * interval + 0.25);
360 constraint_point[k] = 1. * coord_int / interval;
361
362 // The following lines of code should eliminate the problems with
363 // the constraints object which appeared for P>=4. The
364 // AffineConstraints class complained about different constraints
365 // for the same entry: Actually, this
366 // difference could be attributed to FP errors, as it was in the
367 // range of 1.0e-16. These errors originate in the loss of
368 // symmetry in the FP approximation of the shape-functions.
369 // Considering a 3rd order shape function in 1d, we have
370 // N0(x)=N3(1-x) and N1(x)=N2(1-x). For higher order polynomials
371 // the FP approximations of the shape functions do not satisfy
372 // these equations any more! Thus in the following code
373 // everything is computed in the interval x \in [0..0.5], which is
374 // sufficient to express all values that could come out from a
375 // computation of any shape function in the full interval
376 // [0..1]. If x > 0.5 the computation is done for 1-x with the
377 // shape function N_{p-n} instead of N_n. Hence symmetry is
378 // preserved and everything works fine...
379 //
380 // For a different explanation of the problem, see the discussion
381 // in the FiniteElement class for constraint matrices in 3d.
382 mirror[k] = (constraint_point[k] > 0.5);
383 if (mirror[k])
384 constraint_point[k] = 1.0 - constraint_point[k];
385 }
386
387 for (unsigned int j = 0; j < pnts; ++j)
388 {
389 unsigned int indices[2] = {j % (q_deg + 1), j / (q_deg + 1)};
390
391 for (unsigned int k = 0; k < 2; ++k)
392 if (mirror[k])
393 indices[k] = q_deg - indices[k];
394
395 const unsigned int new_index =
396 indices[1] * (q_deg + 1) + indices[0];
397
398 fe.interface_constraints(i, face_index_map[j]) =
399 fe.poly_space->compute_value(index_map_inverse[new_index],
400 constraint_point);
401
402 // if the value is small up to round-off, then simply set it to
403 // zero to avoid unwanted fill-in of the constraint matrices
404 // (which would then increase the number of other DoFs a
405 // constrained DoF would couple to)
406 if (std::fabs(fe.interface_constraints(i, face_index_map[j])) <
407 1e-13)
408 fe.interface_constraints(i, face_index_map[j]) = 0;
409 }
410 }
411 }
412};
413
414#ifndef DOXYGEN
415
416template <int dim, int spacedim>
418 const ScalarPolynomialsBase<dim> &poly_space,
419 const FiniteElementData<dim> &fe_data,
420 const std::vector<bool> &restriction_is_additive_flags)
421 : FE_Poly<dim, spacedim>(
422 poly_space,
423 fe_data,
424 restriction_is_additive_flags,
425 std::vector<ComponentMask>(1, ComponentMask(std::vector<bool>(1, true))))
426 , q_degree(dynamic_cast<const TensorProductPolynomialsBubbles<dim> *>(
427 &poly_space) != nullptr ?
428 this->degree - 1 :
429 this->degree)
430{}
431
432
433
434template <int dim, int spacedim>
435void
436FE_Q_Base<dim, spacedim>::initialize(const std::vector<Point<1>> &points)
437{
438 Assert(points[0][0] == 0,
439 ExcMessage("The first support point has to be zero."));
440 Assert(points.back()[0] == 1,
441 ExcMessage("The last support point has to be one."));
442
443 // distinguish q/q_dg0 case: need to be flexible enough to allow more
444 // degrees of freedom than there are FE_Q degrees of freedom for derived
445 // class FE_Q_DG0 that otherwise shares 95% of the code.
446 const unsigned int q_dofs_per_cell =
447 Utilities::fixed_power<dim>(q_degree + 1);
448 Assert(q_dofs_per_cell == this->n_dofs_per_cell() ||
449 q_dofs_per_cell + 1 == this->n_dofs_per_cell() ||
450 q_dofs_per_cell + dim == this->n_dofs_per_cell(),
452
453 [this, q_dofs_per_cell]() {
454 std::vector<unsigned int> renumber =
455 FETools::hierarchic_to_lexicographic_numbering<dim>(q_degree);
456 for (unsigned int i = q_dofs_per_cell; i < this->n_dofs_per_cell(); ++i)
457 renumber.push_back(i);
458 auto *tensor_poly_space_ptr =
459 dynamic_cast<TensorProductPolynomials<dim> *>(this->poly_space.get());
460 if (tensor_poly_space_ptr != nullptr)
461 {
462 tensor_poly_space_ptr->set_numbering(renumber);
463 return;
464 }
465 auto *tensor_piecewise_poly_space_ptr = dynamic_cast<
467 *>(this->poly_space.get());
468 if (tensor_piecewise_poly_space_ptr != nullptr)
469 {
470 tensor_piecewise_poly_space_ptr->set_numbering(renumber);
471 return;
472 }
473 auto *tensor_bubbles_poly_space_ptr =
475 this->poly_space.get());
476 if (tensor_bubbles_poly_space_ptr != nullptr)
477 {
478 tensor_bubbles_poly_space_ptr->set_numbering(renumber);
479 return;
480 }
481 auto *tensor_const_poly_space_ptr =
483 this->poly_space.get());
484 if (tensor_const_poly_space_ptr != nullptr)
485 {
486 tensor_const_poly_space_ptr->set_numbering(renumber);
487 return;
488 }
490 }();
491
492 // Finally fill in support points on cell and face and initialize
493 // constraints. All of this can happen in parallel
495 tasks += Threads::new_task([&]() { initialize_unit_support_points(points); });
496 tasks +=
498 tasks += Threads::new_task([&]() { initialize_constraints(points); });
499 tasks +=
501 tasks.join_all();
502
503 // do not initialize embedding and restriction here. these matrices are
504 // initialized on demand in get_restriction_matrix and
505 // get_prolongation_matrix
506}
507
508
509
510template <int dim, int spacedim>
511void
513 const FiniteElement<dim, spacedim> &x_source_fe,
514 FullMatrix<double> &interpolation_matrix) const
515{
516 // go through the list of elements we can interpolate from
517 if (const FE_Q_Base<dim, spacedim> *source_fe =
518 dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&x_source_fe))
519 {
520 // ok, source is a Q element, so we will be able to do the work
521 Assert(interpolation_matrix.m() == this->n_dofs_per_cell(),
522 ExcDimensionMismatch(interpolation_matrix.m(),
523 this->n_dofs_per_cell()));
524 Assert(interpolation_matrix.n() == x_source_fe.n_dofs_per_cell(),
525 ExcDimensionMismatch(interpolation_matrix.m(),
526 x_source_fe.n_dofs_per_cell()));
527
528 // only evaluate Q dofs
529 const unsigned int q_dofs_per_cell =
530 Utilities::fixed_power<dim>(q_degree + 1);
531 const unsigned int source_q_dofs_per_cell =
532 Utilities::fixed_power<dim>(source_fe->degree + 1);
533
534 // evaluation is simply done by evaluating the other FE's basis functions
535 // on the unit support points (FE_Q has the property that the cell
536 // interpolation matrix is a unit matrix, so no need to evaluate it and
537 // invert it)
538 for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
539 {
540 // read in a point on this cell and evaluate the shape functions there
541 const Point<dim> p = this->unit_support_points[j];
542
543 // FE_Q element evaluates to 1 in unit support point and to zero in
544 // all other points by construction
545 Assert(std::abs(this->poly_space->compute_value(j, p) - 1.) < 1e-13,
547
548 for (unsigned int i = 0; i < source_q_dofs_per_cell; ++i)
549 interpolation_matrix(j, i) =
550 source_fe->poly_space->compute_value(i, p);
551 }
552
553 // for FE_Q_DG0, add one last row of identity
554 if (q_dofs_per_cell < this->n_dofs_per_cell())
555 {
556 AssertDimension(source_q_dofs_per_cell + 1,
557 source_fe->n_dofs_per_cell());
558 for (unsigned int i = 0; i < source_q_dofs_per_cell; ++i)
559 interpolation_matrix(q_dofs_per_cell, i) = 0.;
560 for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
561 interpolation_matrix(j, source_q_dofs_per_cell) = 0.;
562 interpolation_matrix(q_dofs_per_cell, source_q_dofs_per_cell) = 1.;
563 }
564
565 // cut off very small values
566 const double eps = 2e-13 * q_degree * dim;
567 for (unsigned int i = 0; i < this->n_dofs_per_cell(); ++i)
568 for (unsigned int j = 0; j < source_fe->n_dofs_per_cell(); ++j)
569 if (std::fabs(interpolation_matrix(i, j)) < eps)
570 interpolation_matrix(i, j) = 0.;
571
572# ifdef DEBUG
573 // make sure that the row sum of each of the matrices is 1 at this
574 // point. this must be so since the shape functions sum up to 1
575 for (unsigned int i = 0; i < this->n_dofs_per_cell(); ++i)
576 {
577 double sum = 0.;
578 for (unsigned int j = 0; j < source_fe->n_dofs_per_cell(); ++j)
579 sum += interpolation_matrix(i, j);
580
581 Assert(std::fabs(sum - 1) < eps, ExcInternalError());
582 }
583# endif
584 }
585 else if (dynamic_cast<const FE_Nothing<dim> *>(&x_source_fe))
586 {
587 // the element we want to interpolate from is an FE_Nothing. this
588 // element represents a function that is constant zero and has no
589 // degrees of freedom, so the interpolation is simply a multiplication
590 // with a n_dofs x 0 matrix. there is nothing to do here
591
592 // we would like to verify that the number of rows and columns of
593 // the matrix equals this->n_dofs_per_cell() and zero. unfortunately,
594 // whenever we do FullMatrix::reinit(m,0), it sets both rows and
595 // columns to zero, instead of m and zero. thus, only test the
596 // number of columns
597 Assert(interpolation_matrix.n() == x_source_fe.n_dofs_per_cell(),
598 ExcDimensionMismatch(interpolation_matrix.m(),
599 x_source_fe.n_dofs_per_cell()));
600 }
601 else
603 false,
604 (typename FiniteElement<dim,
606}
607
608
609
610template <int dim, int spacedim>
611void
613 const FiniteElement<dim, spacedim> &source_fe,
614 FullMatrix<double> &interpolation_matrix,
615 const unsigned int face_no) const
616{
619 interpolation_matrix,
620 face_no);
621}
622
623
624
625template <int dim, int spacedim>
626void
628 const FiniteElement<dim, spacedim> &source_fe,
629 const unsigned int subface,
630 FullMatrix<double> &interpolation_matrix,
631 const unsigned int face_no) const
632{
633 Assert(interpolation_matrix.m() == source_fe.n_dofs_per_face(face_no),
634 ExcDimensionMismatch(interpolation_matrix.m(),
635 source_fe.n_dofs_per_face(face_no)));
636
637 Assert(source_fe.n_components() == this->n_components(),
638 ExcDimensionMismatch(source_fe.n_components(), this->n_components()));
639
640 if (source_fe.has_face_support_points(face_no))
641 {
642 // have this test in here since a table of size 2x0 reports its size as
643 // 0x0
644 Assert(interpolation_matrix.n() == this->n_dofs_per_face(face_no),
645 ExcDimensionMismatch(interpolation_matrix.n(),
646 this->n_dofs_per_face(face_no)));
647
648 // Make sure that the element for which the DoFs should be constrained
649 // is the one with the higher polynomial degree. Actually the procedure
650 // will work also if this assertion is not satisfied. But the matrices
651 // produced in that case might lead to problems in the hp-procedures,
652 // which use this method.
653 Assert(
654 this->n_dofs_per_face(face_no) <= source_fe.n_dofs_per_face(face_no),
655 (typename FiniteElement<dim,
657
658 // generate a point on this cell and evaluate the shape functions there
659 const Quadrature<dim - 1> quad_face_support(
660 source_fe.get_unit_face_support_points(face_no));
661
662 // Rule of thumb for FP accuracy, that can be expected for a given
663 // polynomial degree. This value is used to cut off values close to
664 // zero.
665 const double eps = 2e-13 * this->q_degree * std::max(dim - 1, 1);
666
667 // compute the interpolation matrix by simply taking the value at the
668 // support points.
669 // TODO: Verify that all faces are the same with respect to
670 // these support points. Furthermore, check if something has to
671 // be done for the face orientation flag in 3d.
672 const Quadrature<dim> subface_quadrature =
675 quad_face_support,
676 0) :
677 QProjector<dim>::project_to_subface(this->reference_cell(),
678 quad_face_support,
679 0,
680 subface);
681 for (unsigned int i = 0; i < source_fe.n_dofs_per_face(face_no); ++i)
682 {
683 const Point<dim> &p = subface_quadrature.point(i);
684
685 for (unsigned int j = 0; j < this->n_dofs_per_face(face_no); ++j)
686 {
687 double matrix_entry =
688 this->shape_value(this->face_to_cell_index(j, 0), p);
689
690 // Correct the interpolated value. I.e. if it is close to 1 or
691 // 0, make it exactly 1 or 0. Unfortunately, this is required to
692 // avoid problems with higher order elements.
693 if (std::fabs(matrix_entry - 1.0) < eps)
694 matrix_entry = 1.0;
695 if (std::fabs(matrix_entry) < eps)
696 matrix_entry = 0.0;
697
698 interpolation_matrix(i, j) = matrix_entry;
699 }
700 }
701
702# ifdef DEBUG
703 // make sure that the row sum of each of the matrices is 1 at this
704 // point. this must be so since the shape functions sum up to 1
705 for (unsigned int j = 0; j < source_fe.n_dofs_per_face(face_no); ++j)
706 {
707 double sum = 0.;
708
709 for (unsigned int i = 0; i < this->n_dofs_per_face(face_no); ++i)
710 sum += interpolation_matrix(j, i);
711
712 Assert(std::fabs(sum - 1) < eps, ExcInternalError());
713 }
714# endif
715 }
716 else if (dynamic_cast<const FE_Nothing<dim> *>(&source_fe) != nullptr)
717 {
718 // nothing to do here, the FE_Nothing has no degrees of freedom anyway
719 }
720 else
722 false,
723 (typename FiniteElement<dim,
725}
726
727
728
729template <int dim, int spacedim>
730bool
732{
733 return true;
734}
735
736
737
738template <int dim, int spacedim>
739std::vector<std::pair<unsigned int, unsigned int>>
741 const FiniteElement<dim, spacedim> &fe_other) const
742{
743 if (dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other) != nullptr)
744 {
745 // there should be exactly one single DoF of each FE at a vertex, and they
746 // should have identical value
747 return {{0U, 0U}};
748 }
749 else if (dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other) !=
750 nullptr)
751 {
752 // there should be exactly one single DoF of each FE at a vertex, and they
753 // should have identical value
754 return {{0U, 0U}};
755 }
756 else if (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe_other) !=
757 nullptr)
758 {
759 // FE_Hermite will usually have several degrees of freedom on
760 // each vertex, however only the first one will actually
761 // correspond to the shape value at the vertex, meaning it's
762 // the only one of interest for FE_Q_Base
763 return {{0U, 0U}};
764 }
765 else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
766 {
767 // the FE_Nothing has no degrees of freedom, so there are no
768 // equivalencies to be recorded
769 return {};
770 }
771 else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
772 {
773 // if the other element has no elements on faces at all,
774 // then it would be impossible to enforce any kind of
775 // continuity even if we knew exactly what kind of element
776 // we have -- simply because the other element declares
777 // that it is discontinuous because it has no DoFs on
778 // its faces. in that case, just state that we have no
779 // constraints to declare
780 return {};
781 }
782 else
783 {
785 return {};
786 }
787}
788
789
790
791template <int dim, int spacedim>
792std::vector<std::pair<unsigned int, unsigned int>>
794 const FiniteElement<dim, spacedim> &fe_other) const
795{
796 // we can presently only compute these identities if both FEs are FE_Qs or
797 // if the other one is an FE_Nothing
798 if (const FE_Q_Base<dim, spacedim> *fe_q_other =
799 dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other))
800 {
801 // dofs are located along lines, so two dofs are identical if they are
802 // located at identical positions. if we had only equidistant points, we
803 // could simply check for similarity like (i+1)*q == (j+1)*p, but we
804 // might have other support points (e.g. Gauss-Lobatto
805 // points). Therefore, read the points in unit_support_points for the
806 // first coordinate direction. We take the lexicographic ordering of the
807 // points in the first direction (i.e., x-direction), which we access
808 // between index 1 and p-1 (index 0 and p are vertex dofs).
809 const unsigned int p = this->degree;
810 const unsigned int q = fe_q_other->degree;
811
812 std::vector<std::pair<unsigned int, unsigned int>> identities;
813
814 const std::vector<unsigned int> &index_map_inverse =
816 const std::vector<unsigned int> &index_map_inverse_other =
817 fe_q_other->get_poly_space_numbering_inverse();
818
819 for (unsigned int i = 0; i < p - 1; ++i)
820 for (unsigned int j = 0; j < q - 1; ++j)
821 if (std::fabs(
822 this->unit_support_points[index_map_inverse[i + 1]][0] -
823 fe_q_other->unit_support_points[index_map_inverse_other[j + 1]]
824 [0]) < 1e-14)
825 identities.emplace_back(i, j);
826
827 return identities;
828 }
829 else if (const FE_SimplexP<dim, spacedim> *fe_p_other =
830 dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other))
831 {
832 // DoFs are located along lines, so two dofs are identical if they are
833 // located at identical positions. If we had only equidistant points, we
834 // could simply check for similarity like (i+1)*q == (j+1)*p, but we
835 // might have other support points (e.g. Gauss-Lobatto
836 // points). Therefore, read the points in unit_support_points for the
837 // first coordinate direction. For FE_Q, we take the lexicographic
838 // ordering of the line support points in the first direction (i.e.,
839 // x-direction), which we access between index 1 and p-1 (index 0 and p
840 // are vertex dofs). For FE_SimplexP, they are currently hard-coded and we
841 // iterate over points on the first line which begin after the 3 vertex
842 // points in the complete list of unit support points
843
844 Assert(fe_p_other->degree <= 2, ExcNotImplemented());
845
846 const std::vector<unsigned int> &index_map_inverse_q =
848
849 std::vector<std::pair<unsigned int, unsigned int>> identities;
850
851 for (unsigned int i = 0; i < this->degree - 1; ++i)
852 for (unsigned int j = 0; j < fe_p_other->degree - 1; ++j)
853 if (std::fabs(
854 this->unit_support_points[index_map_inverse_q[i + 1]][0] -
855 fe_p_other->get_unit_support_points()[j + 3][0]) < 1e-14)
856 identities.emplace_back(i, j);
857
858 return identities;
859 }
860 else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
861 {
862 // the FE_Nothing has no degrees of freedom, so there are no
863 // equivalencies to be recorded
864 return {};
865 }
866 else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
867 {
868 // if the other element has no elements on faces at all,
869 // then it would be impossible to enforce any kind of
870 // continuity even if we knew exactly what kind of element
871 // we have -- simply because the other element declares
872 // that it is discontinuous because it has no DoFs on
873 // its faces. in that case, just state that we have no
874 // constraints to declare
875 return {};
876 }
877 else
878 {
880 return {};
881 }
882}
883
884
885
886template <int dim, int spacedim>
887std::vector<std::pair<unsigned int, unsigned int>>
889 const FiniteElement<dim, spacedim> &fe_other,
890 const unsigned int) const
891{
892 // we can presently only compute these identities if both FEs are FE_Qs or
893 // if the other one is an FE_Nothing
894 if (const FE_Q_Base<dim, spacedim> *fe_q_other =
895 dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other))
896 {
897 // this works exactly like the line case above, except that now we have
898 // to have two indices i1, i2 and j1, j2 to characterize the dofs on the
899 // face of each of the finite elements. since they are ordered
900 // lexicographically along the first line and we have a tensor product,
901 // the rest is rather straightforward
902 const unsigned int p = this->degree;
903 const unsigned int q = fe_q_other->degree;
904
905 std::vector<std::pair<unsigned int, unsigned int>> identities;
906
907 const std::vector<unsigned int> &index_map_inverse =
909 const std::vector<unsigned int> &index_map_inverse_other =
910 fe_q_other->get_poly_space_numbering_inverse();
911
912 for (unsigned int i1 = 0; i1 < p - 1; ++i1)
913 for (unsigned int i2 = 0; i2 < p - 1; ++i2)
914 for (unsigned int j1 = 0; j1 < q - 1; ++j1)
915 for (unsigned int j2 = 0; j2 < q - 1; ++j2)
916 if ((std::fabs(
917 this->unit_support_points[index_map_inverse[i1 + 1]][0] -
918 fe_q_other
919 ->unit_support_points[index_map_inverse_other[j1 + 1]]
920 [0]) < 1e-14) &&
921 (std::fabs(
922 this->unit_support_points[index_map_inverse[i2 + 1]][0] -
923 fe_q_other
924 ->unit_support_points[index_map_inverse_other[j2 + 1]]
925 [0]) < 1e-14))
926 identities.emplace_back(i1 * (p - 1) + i2, j1 * (q - 1) + j2);
927
928 return identities;
929 }
930 else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
931 {
932 // the FE_Nothing has no degrees of freedom, so there are no
933 // equivalencies to be recorded
934 return std::vector<std::pair<unsigned int, unsigned int>>();
935 }
936 else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
937 {
938 // if the other element has no elements on faces at all,
939 // then it would be impossible to enforce any kind of
940 // continuity even if we knew exactly what kind of element
941 // we have -- simply because the other element declares
942 // that it is discontinuous because it has no DoFs on
943 // its faces. in that case, just state that we have no
944 // constraints to declare
945 return std::vector<std::pair<unsigned int, unsigned int>>();
946 }
947 else
948 {
950 return std::vector<std::pair<unsigned int, unsigned int>>();
951 }
952}
953
954
955
956//---------------------------------------------------------------------------
957// Auxiliary functions
958//---------------------------------------------------------------------------
959
960
961
962template <int dim, int spacedim>
963void
965 const std::vector<Point<1>> &points)
966{
967 const std::vector<unsigned int> &index_map_inverse =
969
970 // We can compute the support points by computing the tensor
971 // product of the 1d set of points. We could do this by hand, but it's
972 // easier to just re-use functionality that's already been implemented
973 // for quadrature formulas.
974 const Quadrature<1> support_1d(points);
975 const Quadrature<dim> support_quadrature(support_1d); // NOLINT
976
977 // The only thing we have to do is reorder the points from tensor
978 // product order to the order in which we enumerate DoFs on cells
979 this->unit_support_points.resize(support_quadrature.size());
980 for (unsigned int k = 0; k < support_quadrature.size(); ++k)
981 this->unit_support_points[index_map_inverse[k]] =
982 support_quadrature.point(k);
983}
984
985
986
987template <int dim, int spacedim>
988void
990 const std::vector<Point<1>> &points)
991{
992 // TODO: the implementation makes the assumption that all faces have the
993 // same number of dofs
995 const unsigned int face_no = 0;
996
997 this->unit_face_support_points[face_no].resize(
998 Utilities::fixed_power<dim - 1>(q_degree + 1));
999
1000 // In 1d, there is only one 0-dimensional support point, so there is nothing
1001 // more to be done.
1002 if (dim == 1)
1003 return;
1004
1005 // find renumbering of faces and assign from values of quadrature
1006 const std::vector<unsigned int> face_index_map =
1008
1009 // We can compute the support points by computing the tensor
1010 // product of the 1d set of points. We could do this by hand, but it's
1011 // easier to just re-use functionality that's already been implemented
1012 // for quadrature formulas.
1013 const Quadrature<1> support_1d(points);
1014 const Quadrature<dim - 1> support_quadrature(support_1d); // NOLINT
1015
1016 // The only thing we have to do is reorder the points from tensor
1017 // product order to the order in which we enumerate DoFs on cells
1018 this->unit_face_support_points[face_no].resize(support_quadrature.size());
1019 for (unsigned int k = 0; k < support_quadrature.size(); ++k)
1020 this->unit_face_support_points[face_no][face_index_map[k]] =
1021 support_quadrature.point(k);
1022}
1023
1024
1025
1026template <int dim, int spacedim>
1027void
1029{
1030 // initialize reordering of line dofs
1031 for (unsigned int i = 0; i < this->n_dofs_per_line(); ++i)
1033 this->n_dofs_per_line() - 1 - i - i;
1034
1035 // for 1d and 2d we can skip adjust_quad_dof_index_for_face_orientation_table
1036 if (dim < 3)
1037 return;
1038
1039 // TODO: the implementation makes the assumption that all faces have the
1040 // same number of dofs
1041 AssertDimension(this->n_unique_faces(), 1);
1042 const unsigned int face_no = 0;
1043
1044 Assert(
1046 this->reference_cell().n_face_orientations(face_no) *
1047 this->n_dofs_per_quad(face_no),
1049
1050 const unsigned int n = q_degree - 1;
1051 Assert(n * n == this->n_dofs_per_quad(face_no), ExcInternalError());
1052
1053 // the dofs on a face are connected to a n x n matrix. for example, for
1054 // degree==4 we have the following dofs on a quad
1055
1056 // ___________
1057 // | |
1058 // | 6 7 8 |
1059 // | |
1060 // | 3 4 5 |
1061 // | |
1062 // | 0 1 2 |
1063 // |___________|
1064 //
1065 // we have dof_no=i+n*j with index i in x-direction and index j in
1066 // y-direction running from 0 to n-1. to extract i and j we can use
1067 // i=dof_no%n and j=dof_no/n. i and j can then be used to construct the
1068 // rotated and mirrored numbers.
1069
1070
1071 for (unsigned int local = 0; local < this->n_dofs_per_quad(face_no); ++local)
1072 // face support points are in lexicographic ordering with x running
1073 // fastest. invert that (y running fastest)
1074 {
1075 unsigned int i = local % n, j = local / n;
1076
1077 // face_orientation=false, face_flip=false, face_rotation=false
1079 local, internal::combined_face_orientation(false, false, false)) =
1080 j + i * n - local;
1081 // face_orientation=false, face_flip=false, face_rotation=true
1083 local, internal::combined_face_orientation(false, true, false)) =
1084 i + (n - 1 - j) * n - local;
1085 // face_orientation=false, face_flip=true, face_rotation=false
1087 local, internal::combined_face_orientation(false, false, true)) =
1088 (n - 1 - j) + (n - 1 - i) * n - local;
1089 // face_orientation=false, face_flip=true, face_rotation=true
1091 local, internal::combined_face_orientation(false, true, true)) =
1092 (n - 1 - i) + j * n - local;
1093 // face_orientation=true, face_flip=false, face_rotation=false
1095 local, internal::combined_face_orientation(true, false, false)) = 0;
1096 // face_orientation=true, face_flip=false, face_rotation=true
1098 local, internal::combined_face_orientation(true, true, false)) =
1099 j + (n - 1 - i) * n - local;
1100 // face_orientation=true, face_flip=true, face_rotation=false
1102 local, internal::combined_face_orientation(true, false, true)) =
1103 (n - 1 - i) + (n - 1 - j) * n - local;
1104 // face_orientation=true, face_flip=true, face_rotation=true
1106 local, internal::combined_face_orientation(true, true, true)) =
1107 (n - 1 - j) + i * n - local;
1108 }
1109}
1110
1111
1112
1113template <int dim, int spacedim>
1114unsigned int
1116 const unsigned int face_index,
1117 const unsigned int face,
1118 const unsigned char combined_orientation) const
1119{
1120 AssertIndexRange(face_index, this->n_dofs_per_face(face));
1122
1123 // we need to distinguish between DoFs on vertices, lines and in 3d quads.
1124 // do so in a sequence of if-else statements
1125 if (face_index < this->get_first_face_line_index(face))
1126 // DoF is on a vertex
1127 {
1128 // get the number of the vertex on the face that corresponds to this DoF,
1129 // along with the number of the DoF on this vertex
1130 const unsigned int face_vertex = face_index / this->n_dofs_per_vertex();
1131 const unsigned int dof_index_on_vertex =
1132 face_index % this->n_dofs_per_vertex();
1133
1134 // then get the number of this vertex on the cell and translate
1135 // this to a DoF number on the cell
1137 face, face_vertex, combined_orientation) *
1138 this->n_dofs_per_vertex() +
1139 dof_index_on_vertex;
1140 }
1141 else if (face_index < this->get_first_face_quad_index(face))
1142 // DoF is on a face
1143 {
1144 // do the same kind of translation as before. we need to only consider
1145 // DoFs on the lines, i.e., ignoring those on the vertices
1146 const unsigned int index =
1147 face_index - this->get_first_face_line_index(face);
1148
1149 const unsigned int face_line = index / this->n_dofs_per_line();
1150 const unsigned int dof_index_on_line = index % this->n_dofs_per_line();
1151
1152 // we now also need to adjust the line index for the case of
1153 // face orientation, face flips, etc
1154 unsigned int adjusted_dof_index_on_line = 0;
1155 switch (dim)
1156 {
1157 case 1:
1159 break;
1160
1161 case 2:
1162 if (combined_orientation ==
1164 adjusted_dof_index_on_line = dof_index_on_line;
1165 else
1166 adjusted_dof_index_on_line =
1167 this->n_dofs_per_line() - dof_index_on_line - 1;
1168 break;
1169
1170 case 3:
1171 // in 3d, things are difficult. someone will have to think
1172 // about how this code here should look like, by drawing a bunch
1173 // of pictures of how all the faces can look like with the various
1174 // flips and rotations.
1175 //
1176 // that said, the Q2 case is easy enough to implement, as is the
1177 // case where everything is in standard orientation
1178 Assert((this->n_dofs_per_line() <= 1) ||
1179 combined_orientation ==
1182 adjusted_dof_index_on_line = dof_index_on_line;
1183 break;
1184
1185 default:
1187 }
1188
1189 return (this->get_first_line_index() +
1190 this->reference_cell().face_to_cell_lines(face,
1191 face_line,
1192 combined_orientation) *
1193 this->n_dofs_per_line() +
1194 adjusted_dof_index_on_line);
1195 }
1196 else
1197 // DoF is on a quad
1198 {
1199 Assert(dim >= 3, ExcInternalError());
1200
1201 // ignore vertex and line dofs
1202 const unsigned int index =
1203 face_index - this->get_first_face_quad_index(face);
1204
1205 // the same is true here as above for the 3d case -- someone will
1206 // just have to draw a bunch of pictures. in the meantime,
1207 // we can implement the Q2 case in which it is simple
1208 Assert((this->n_dofs_per_quad(face) <= 1) ||
1209 combined_orientation ==
1212 return (this->get_first_quad_index(face) + index);
1213 }
1214}
1215
1216
1217
1218template <int dim, int spacedim>
1219std::vector<unsigned int>
1221{
1222 using FEQ = FE_Q_Base<dim, spacedim>;
1223 AssertThrow(degree > 0, typename FEQ::ExcFEQCannotHaveDegree0());
1224 std::vector<unsigned int> dpo(dim + 1, 1U);
1225 for (unsigned int i = 1; i < dpo.size(); ++i)
1226 dpo[i] = dpo[i - 1] * (degree - 1);
1227 return dpo;
1228}
1229
1230
1231
1232template <int dim, int spacedim>
1233void
1235 const std::vector<Point<1>> &points)
1236{
1238}
1239
1240
1241
1242template <int dim, int spacedim>
1243const FullMatrix<double> &
1245 const unsigned int child,
1246 const RefinementCase<dim> &refinement_case) const
1247{
1248 AssertIndexRange(refinement_case,
1250 Assert(refinement_case != RefinementCase<dim>::no_refinement,
1251 ExcMessage(
1252 "Prolongation matrices are only available for refined cells!"));
1253 AssertIndexRange(child, GeometryInfo<dim>::n_children(refinement_case));
1254
1255 // initialization upon first request
1256 if (this->prolongation[refinement_case - 1][child].n() == 0)
1257 {
1258 std::lock_guard<std::mutex> lock(prolongation_matrix_mutex);
1259
1260 // if matrix got updated while waiting for the lock
1261 if (this->prolongation[refinement_case - 1][child].n() ==
1262 this->n_dofs_per_cell())
1263 return this->prolongation[refinement_case - 1][child];
1264
1265 // distinguish q/q_dg0 case: only treat Q dofs first
1266 const unsigned int q_dofs_per_cell =
1267 Utilities::fixed_power<dim>(q_degree + 1);
1268
1269 // compute the interpolation matrices in much the same way as we do for
1270 // the constraints. it's actually simpler here, since we don't have this
1271 // weird renumbering stuff going on. The trick is again that we the
1272 // interpolation matrix is formed by a permutation of the indices of the
1273 // cell matrix. The value eps is used a threshold to decide when certain
1274 // evaluations of the Lagrange polynomials are zero or one.
1275 const double eps = 1e-15 * q_degree * dim;
1276
1277# ifdef DEBUG
1278 // in DEBUG mode, check that the evaluation of support points in the
1279 // current numbering gives the identity operation
1280 for (unsigned int i = 0; i < q_dofs_per_cell; ++i)
1281 {
1282 Assert(std::fabs(1. - this->poly_space->compute_value(
1283 i, this->unit_support_points[i])) < eps,
1284 ExcInternalError("The Lagrange polynomial does not evaluate "
1285 "to one or zero in a nodal point. "
1286 "This typically indicates that the "
1287 "polynomial interpolation is "
1288 "ill-conditioned such that round-off "
1289 "prevents the sum to be one."));
1290 for (unsigned int j = 0; j < q_dofs_per_cell; ++j)
1291 if (j != i)
1292 Assert(std::fabs(this->poly_space->compute_value(
1293 i, this->unit_support_points[j])) < eps,
1295 "The Lagrange polynomial does not evaluate "
1296 "to one or zero in a nodal point. "
1297 "This typically indicates that the "
1298 "polynomial interpolation is "
1299 "ill-conditioned such that round-off "
1300 "prevents the sum to be one."));
1301 }
1302# endif
1303
1304 // to efficiently evaluate the polynomial at the subcell, make use of
1305 // the tensor product structure of this element and only evaluate 1d
1306 // information from the polynomial. This makes the cost of this function
1307 // almost negligible also for high order elements
1308 const unsigned int dofs1d = q_degree + 1;
1309 std::vector<Table<2, double>> subcell_evaluations(
1310 dim, Table<2, double>(dofs1d, dofs1d));
1311
1312 const std::vector<unsigned int> &index_map_inverse =
1314
1315 // helper value: step size how to walk through diagonal and how many
1316 // points we have left apart from the first dimension
1317 unsigned int step_size_diag = 0;
1318 {
1319 unsigned int factor = 1;
1320 for (unsigned int d = 0; d < dim; ++d)
1321 {
1322 step_size_diag += factor;
1323 factor *= dofs1d;
1324 }
1325 }
1326
1327 FullMatrix<double> prolongate(this->n_dofs_per_cell(),
1328 this->n_dofs_per_cell());
1329
1330 // go through the points in diagonal to capture variation in all
1331 // directions simultaneously
1332 for (unsigned int j = 0; j < dofs1d; ++j)
1333 {
1334 const unsigned int diag_comp = index_map_inverse[j * step_size_diag];
1335 const Point<dim> p_subcell = this->unit_support_points[diag_comp];
1336 const Point<dim> p_cell =
1338 child,
1339 refinement_case);
1340 for (unsigned int i = 0; i < dofs1d; ++i)
1341 for (unsigned int d = 0; d < dim; ++d)
1342 {
1343 // evaluate along line where only x is different from zero
1345 point[0] = p_cell[d];
1346 const double cell_value =
1347 this->poly_space->compute_value(index_map_inverse[i], point);
1348
1349 // cut off values that are too small. note that we have here
1350 // Lagrange interpolation functions, so they should be zero at
1351 // almost all points, and one at the others, at least on the
1352 // subcells. so set them to their exact values
1353 //
1354 // the actual cut-off value is somewhat fuzzy, but it works
1355 // for 2e-13*degree*dim (see above), which is kind of
1356 // reasonable given that we compute the values of the
1357 // polynomials via an degree-step recursion and then multiply
1358 // the 1d-values. this gives us a linear growth in degree*dim,
1359 // times a small constant.
1360 //
1361 // the embedding matrix is given by applying the inverse of
1362 // the subcell matrix on the cell_interpolation matrix. since
1363 // the subcell matrix is actually only a permutation vector,
1364 // all we need to do is to switch the rows we write the data
1365 // into. moreover, cut off very small values here
1366 if (std::fabs(cell_value) < eps)
1367 subcell_evaluations[d](j, i) = 0;
1368 else
1369 subcell_evaluations[d](j, i) = cell_value;
1370 }
1371 }
1372
1373 // now expand from 1d info. block innermost dimension (x_0) in order to
1374 // avoid difficult checks at innermost loop
1375 unsigned int j_indices[dim];
1376 internal::FE_Q_Base::zero_indices<dim>(j_indices);
1377 for (unsigned int j = 0; j < q_dofs_per_cell; j += dofs1d)
1378 {
1379 unsigned int i_indices[dim];
1380 internal::FE_Q_Base::zero_indices<dim>(i_indices);
1381 for (unsigned int i = 0; i < q_dofs_per_cell; i += dofs1d)
1382 {
1383 double val_extra_dim = 1.;
1384 for (unsigned int d = 1; d < dim; ++d)
1385 val_extra_dim *=
1386 subcell_evaluations[d](j_indices[d - 1], i_indices[d - 1]);
1387
1388 // innermost sum where we actually compute. the same as
1389 // prolongate(j,i) = this->poly_space->compute_value (i, p_cell)
1390 for (unsigned int jj = 0; jj < dofs1d; ++jj)
1391 {
1392 const unsigned int j_ind = index_map_inverse[j + jj];
1393 for (unsigned int ii = 0; ii < dofs1d; ++ii)
1394 prolongate(j_ind, index_map_inverse[i + ii]) =
1395 val_extra_dim * subcell_evaluations[0](jj, ii);
1396 }
1397
1398 // update indices that denote the tensor product position. a bit
1399 // fuzzy and therefore not done for innermost x_0 direction
1400 internal::FE_Q_Base::increment_indices<dim>(i_indices, dofs1d);
1401 }
1402 Assert(i_indices[dim - 1] == 1, ExcInternalError());
1403 internal::FE_Q_Base::increment_indices<dim>(j_indices, dofs1d);
1404 }
1405
1406 // the discontinuous node is simply mapped on the discontinuous node on
1407 // the child element
1408 if (q_dofs_per_cell < this->n_dofs_per_cell())
1409 prolongate(q_dofs_per_cell, q_dofs_per_cell) = 1.;
1410
1411 // and make sure that the row sum is 1. this must be so since for this
1412 // element, the shape functions add up to one
1413# ifdef DEBUG
1414 for (unsigned int row = 0; row < this->n_dofs_per_cell(); ++row)
1415 {
1416 double sum = 0;
1417 for (unsigned int col = 0; col < this->n_dofs_per_cell(); ++col)
1418 sum += prolongate(row, col);
1419 Assert(std::fabs(sum - 1.) <
1420 std::max(eps, 5e-16 * std::sqrt(this->n_dofs_per_cell())),
1421 ExcInternalError("The entries in a row of the local "
1422 "prolongation matrix do not add to one. "
1423 "This typically indicates that the "
1424 "polynomial interpolation is "
1425 "ill-conditioned such that round-off "
1426 "prevents the sum to be one."));
1427 }
1428# endif
1429
1430 // move result into place
1431 const_cast<FullMatrix<double> &>(
1432 this->prolongation[refinement_case - 1][child]) = std::move(prolongate);
1433 }
1434
1435 // finally return the matrix
1436 return this->prolongation[refinement_case - 1][child];
1437}
1438
1439
1440
1441template <int dim, int spacedim>
1442const FullMatrix<double> &
1444 const unsigned int child,
1445 const RefinementCase<dim> &refinement_case) const
1446{
1447 AssertIndexRange(refinement_case,
1449 Assert(refinement_case != RefinementCase<dim>::no_refinement,
1450 ExcMessage(
1451 "Restriction matrices are only available for refined cells!"));
1452 AssertIndexRange(child, GeometryInfo<dim>::n_children(refinement_case));
1453
1454 // initialization upon first request
1455 if (this->restriction[refinement_case - 1][child].n() == 0)
1456 {
1457 std::lock_guard<std::mutex> lock(restriction_matrix_mutex);
1458
1459 // if matrix got updated while waiting for the lock...
1460 if (this->restriction[refinement_case - 1][child].n() ==
1461 this->n_dofs_per_cell())
1462 return this->restriction[refinement_case - 1][child];
1463
1464 FullMatrix<double> my_restriction(this->n_dofs_per_cell(),
1465 this->n_dofs_per_cell());
1466 // distinguish q/q_dg0 case
1467 const unsigned int q_dofs_per_cell =
1468 Utilities::fixed_power<dim>(q_degree + 1);
1469
1470 // for Lagrange interpolation polynomials based on equidistant points,
1471 // construction of the restriction matrices is relatively simple. the
1472 // reason is that in this case the interpolation points on the mother
1473 // cell are always also interpolation points for some shape function on
1474 // one or the other child.
1475 //
1476 // in the general case with non-equidistant points, we need to actually
1477 // do an interpolation. thus, we take the interpolation points on the
1478 // mother cell and evaluate the shape functions of the child cell on
1479 // those points. it does not hurt in the equidistant case because then
1480 // simple one shape function evaluates to one and the others to zero.
1481 //
1482 // this element is non-additive in all its degrees of freedom by
1483 // default, which requires care in downstream use. fortunately, even the
1484 // interpolation on non-equidistant points is invariant under the
1485 // assumption that whenever a row makes a non-zero contribution to the
1486 // mother's residual, the correct value is interpolated.
1487
1488 const double eps = 1e-15 * q_degree * dim;
1489 const std::vector<unsigned int> &index_map_inverse =
1491
1492 const unsigned int dofs1d = q_degree + 1;
1493 std::vector<Tensor<1, dim>> evaluations1d(dofs1d);
1494
1495 my_restriction.reinit(this->n_dofs_per_cell(), this->n_dofs_per_cell());
1496
1497 for (unsigned int i = 0; i < q_dofs_per_cell; ++i)
1498 {
1499 unsigned int mother_dof = index_map_inverse[i];
1500 const Point<dim> p_cell = this->unit_support_points[mother_dof];
1501
1502 // check whether this interpolation point is inside this child cell
1503 const Point<dim> p_subcell =
1505 child,
1506 refinement_case);
1508 {
1509 // same logic as in initialize_embedding to evaluate the
1510 // polynomial faster than from the tensor product: since we
1511 // evaluate all polynomials, it is much faster to just compute
1512 // the 1d values for all polynomials before and then get the
1513 // dim-data.
1514 for (unsigned int j = 0; j < dofs1d; ++j)
1515 for (unsigned int d = 0; d < dim; ++d)
1516 {
1518 point[0] = p_subcell[d];
1519 evaluations1d[j][d] =
1520 this->poly_space->compute_value(index_map_inverse[j],
1521 point);
1522 }
1523 unsigned int j_indices[dim];
1524 internal::FE_Q_Base::zero_indices<dim>(j_indices);
1525 double sum_check = 0;
1526 for (unsigned int j = 0; j < q_dofs_per_cell; j += dofs1d)
1527 {
1528 double val_extra_dim = 1.;
1529 for (unsigned int d = 1; d < dim; ++d)
1530 val_extra_dim *= evaluations1d[j_indices[d - 1]][d];
1531 for (unsigned int jj = 0; jj < dofs1d; ++jj)
1532 {
1533 // find the child shape function(s) corresponding to
1534 // this point. Usually this is just one function;
1535 // however, when we use FE_Q on arbitrary nodes a parent
1536 // support point might not be a child support point, and
1537 // then we will get more than one nonzero value per
1538 // row. Still, the values should sum up to 1
1539 const double val = val_extra_dim * evaluations1d[jj][0];
1540 const unsigned int child_dof = index_map_inverse[j + jj];
1541 if (std::fabs(val - 1.) < eps)
1542 my_restriction(mother_dof, child_dof) = 1.;
1543 else if (std::fabs(val) > eps)
1544 my_restriction(mother_dof, child_dof) = val;
1545 sum_check += val;
1546 }
1547 internal::FE_Q_Base::increment_indices<dim>(j_indices,
1548 dofs1d);
1549 }
1550 (void)sum_check;
1551 Assert(std::fabs(sum_check - 1.0) <
1552 std::max(eps,
1553 5e-16 * std::sqrt(this->n_dofs_per_cell())),
1554 ExcInternalError("The entries in a row of the local "
1555 "restriction matrix do not add to one. "
1556 "This typically indicates that the "
1557 "polynomial interpolation is "
1558 "ill-conditioned such that round-off "
1559 "prevents the sum to be one."));
1560 }
1561
1562 // part for FE_Q_DG0
1563 if (q_dofs_per_cell < this->n_dofs_per_cell())
1564 my_restriction(this->n_dofs_per_cell() - 1,
1565 this->n_dofs_per_cell() - 1) =
1567 RefinementCase<dim>(refinement_case));
1568 }
1569
1570 // move result into place
1571 const_cast<FullMatrix<double> &>(
1572 this->restriction[refinement_case - 1][child]) =
1573 std::move(my_restriction);
1574 }
1575
1576 return this->restriction[refinement_case - 1][child];
1577}
1578
1579
1580
1581//---------------------------------------------------------------------------
1582// Data field initialization
1583//---------------------------------------------------------------------------
1584
1585
1586template <int dim, int spacedim>
1587bool
1589 const unsigned int shape_index,
1590 const unsigned int face_index) const
1591{
1592 AssertIndexRange(shape_index, this->n_dofs_per_cell());
1594
1595 // in 1d, things are simple. since there is only one degree of freedom per
1596 // vertex in this class, the first is on vertex 0 (==face 0 in some sense),
1597 // the second on face 1:
1598 if (dim == 1)
1599 return (((shape_index == 0) && (face_index == 0)) ||
1600 ((shape_index == 1) && (face_index == 1)));
1601
1602 // first, special-case interior shape functions, since they have no support
1603 // no-where on the boundary
1604 if (((dim == 2) &&
1605 (shape_index >= this->get_first_quad_index(0 /*first quad*/))) ||
1606 ((dim == 3) && (shape_index >= this->get_first_hex_index())))
1607 return false;
1608
1609 // let's see whether this is a vertex
1610 if (shape_index < this->get_first_line_index())
1611 {
1612 // for Q elements, there is one dof per vertex, so
1613 // shape_index==vertex_number. check whether this vertex is on the given
1614 // face. thus, for each face, give a list of vertices
1615 const unsigned int vertex_no = shape_index;
1618
1619 for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_face; ++v)
1620 if (GeometryInfo<dim>::face_to_cell_vertices(face_index, v) ==
1621 vertex_no)
1622 return true;
1623
1624 return false;
1625 }
1626 else if (shape_index < this->get_first_quad_index(0 /*first quad*/))
1627 // ok, dof is on a line
1628 {
1629 const unsigned int line_index =
1630 (shape_index - this->get_first_line_index()) / this->n_dofs_per_line();
1633
1634 // in 2d, the line is the face, so get the line index
1635 if constexpr (dim == 2)
1636 return (line_index == face_index);
1637 else if constexpr (dim == 3)
1638 {
1639 // see whether the given line is on the given face.
1640 for (unsigned int l = 0; l < GeometryInfo<3>::lines_per_face; ++l)
1641 if (GeometryInfo<3>::face_to_cell_lines(face_index, l) ==
1642 line_index)
1643 return true;
1644
1645 return false;
1646 }
1647 else
1649 }
1650 else if (shape_index < this->get_first_hex_index())
1651 // dof is on a quad
1652 {
1653 const unsigned int quad_index =
1654 (shape_index - this->get_first_quad_index(0)) /
1655 this->n_dofs_per_quad(face_index); // this won't work
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
1667 }
1668 else
1669 // dof on hex
1670 {
1671 // can only happen in 3d, but this case has already been covered above
1673 return false;
1674 }
1675
1676 // we should not have gotten here
1678 return false;
1679}
1680
1681
1682
1683template <int dim, int spacedim>
1684std::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#endif
1699
1700// explicit instantiations
1701#include "fe_q_base.inst"
1702
std::vector< unsigned int > get_poly_space_numbering_inverse() const
const ScalarPolynomialsBase< dim > & get_poly_space() const
virtual double shape_value(const unsigned int i, const Point< dim > &p) const override
const std::unique_ptr< ScalarPolynomialsBase< dim > > poly_space
Definition fe_poly.h:532
Threads::Mutex prolongation_matrix_mutex
Definition fe_q_base.h:334
void initialize_unit_face_support_points(const std::vector< Point< 1 > > &points)
virtual bool hp_constraints_are_implemented() const override
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_line_dof_identities(const FiniteElement< dim, spacedim > &fe_other) const override
virtual void get_interpolation_matrix(const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const override
Threads::Mutex restriction_matrix_mutex
Definition fe_q_base.h:333
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
const unsigned int q_degree
Definition fe_q_base.h:341
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > get_constant_modes() const override
void initialize(const std::vector< Point< 1 > > &support_points_1d)
void initialize_unit_support_points(const std::vector< Point< 1 > > &points)
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
virtual void get_face_interpolation_matrix(const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix, const unsigned int face_no=0) const override
static std::vector< unsigned int > get_dpo_vector(const unsigned int degree)
void initialize_constraints(const std::vector< Point< 1 > > &points)
void initialize_dof_index_permutations()
virtual const FullMatrix< double > & get_restriction_matrix(const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
virtual const FullMatrix< double > & get_prolongation_matrix(const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
virtual unsigned int face_to_cell_index(const unsigned int face_dof_index, const unsigned int face, const unsigned char combined_orientation=ReferenceCell::default_combined_face_orientation()) const override
virtual bool has_support_on_face(const unsigned int shape_index, const unsigned int face_index) const override
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_vertex_dof_identities(const FiniteElement< dim, spacedim > &fe_other) const override
FE_Q_Base(const ScalarPolynomialsBase< dim > &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags)
unsigned int get_first_line_index() const
unsigned int n_dofs_per_vertex() const
const unsigned int degree
Definition fe_data.h:452
unsigned int n_dofs_per_cell() const
unsigned int n_dofs_per_line() const
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
unsigned int n_components() const
ReferenceCell reference_cell() const
unsigned int get_first_face_line_index(const unsigned int face_no=0) const
unsigned int get_first_face_quad_index(const unsigned int face_no=0) const
unsigned int n_unique_faces() const
unsigned int n_dofs_per_quad(unsigned int face_no=0) const
unsigned int get_first_hex_index() const
std::vector< std::vector< Point< dim - 1 > > > unit_face_support_points
Definition fe.h:2458
bool has_face_support_points(const unsigned int face_no=0) const
std::vector< std::vector< FullMatrix< double > > > restriction
Definition fe.h:2413
std::vector< Table< 2, int > > adjust_quad_dof_index_for_face_orientation_table
Definition fe.h:2487
const std::vector< Point< dim - 1 > > & get_unit_face_support_points(const unsigned int face_no=0) const
std::vector< int > adjust_line_dof_index_for_line_orientation_table
Definition fe.h:2500
TableIndices< 2 > interface_constraints_size() const
std::vector< Point< dim > > unit_support_points
Definition fe.h:2451
FullMatrix< double > interface_constraints
Definition fe.h:2439
std::vector< std::vector< FullMatrix< double > > > prolongation
Definition fe.h:2427
size_type n() const
size_type m() const
Definition point.h:111
static void project_to_subface(const ReferenceCell &reference_cell, 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)
static void project_to_face(const ReferenceCell &reference_cell, const SubQuadrature &quadrature, const unsigned int face_no, std::vector< Point< dim > > &q_points)
const Point< dim > & point(const unsigned int i) const
unsigned int face_to_cell_vertices(const unsigned int face, const unsigned int vertex, const unsigned char face_orientation) const
static constexpr unsigned char default_combined_face_orientation()
void set_numbering(const std::vector< unsigned int > &renumber)
void set_numbering(const std::vector< unsigned int > &renumber)
void set_numbering(const std::vector< unsigned int > &renumber)
#define DEAL_II_NAMESPACE_OPEN
Definition config.h:502
#define DEAL_II_NAMESPACE_CLOSE
Definition config.h:503
static ::ExceptionBase & ExcNotImplemented()
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
#define AssertIndexRange(index, range)
static ::ExceptionBase & ExcInternalError()
static ::ExceptionBase & ExcDimensionMismatch(std::size_t arg1, std::size_t arg2)
static ::ExceptionBase & ExcInterpolationNotImplemented()
static ::ExceptionBase & ExcMessage(std::string arg1)
#define AssertThrow(cond, exc)
Task< RT > new_task(const std::function< RT()> &function)
#define DEAL_II_ASSERT_UNREACHABLE()
#define DEAL_II_NOT_IMPLEMENTED()
std::vector< unsigned int > lexicographic_to_hierarchic_numbering(unsigned int degree)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:191
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned char combined_face_orientation(const bool face_orientation, const bool face_rotation, const bool face_flip)
static const unsigned int invalid_unsigned_int
Definition types.h:220
STL namespace.
::VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
static void initialize_constraints(const std::vector< Point< 1 > > &, FE_Q_Base< 3, spacedim > &fe)
Definition fe_q_base.cc:217
static void initialize_constraints(const std::vector< Point< 1 > > &, FE_Q_Base< 1, spacedim > &)
Definition fe_q_base.cc:97
static void initialize_constraints(const std::vector< Point< 1 > > &, FE_Q_Base< 2, spacedim > &fe)
Definition fe_q_base.cc:106
static unsigned int n_children(const RefinementCase< dim > &refinement_case)
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)
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)