Reference documentation for deal.II version 9.6.0
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step-75.h
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1
467) const override
468 *   {
469 *   const std::array<double, 2> polar =
471 *  
472 *   constexpr const double alpha = 2. / 3.;
473 *   return std::pow(polar[0], alpha) * std::sin(alpha * polar[1]);
474 *   }
475 *   };
476 *  
477 *  
478 *  
479 * @endcode
480 *
481 *
482 * <a name="step_75-Parameters"></a>
483 * <h3>Parameters</h3>
484 *
485
486 *
487 * For this tutorial, we will use a simplified set of parameters. It is also
488 * possible to use a ParameterHandler class here, but to keep this tutorial
489 * short we decided on using simple structs. The actual intention of all these
490 * parameters will be described in the upcoming classes at their respective
491 * location where they are used.
492 *
493
494 *
495 * The following parameter set controls the coarse-grid solver, the smoothers,
496 * and the inter-grid transfer scheme of the multigrid mechanism.
497 * We populate it with default parameters.
498 *
499 * @code
500 *   struct MultigridParameters
501 *   {
502 *   struct
503 *   {
504 *   std::string type = "cg_with_amg";
505 *   unsigned int maxiter = 10000;
506 *   double abstol = 1e-20;
507 *   double reltol = 1e-4;
508 *   unsigned int smoother_sweeps = 1;
509 *   unsigned int n_cycles = 1;
510 *   std::string smoother_type = "ILU";
511 *   } coarse_solver;
512 *  
513 *   struct
514 *   {
515 *   std::string type = "chebyshev";
516 *   double smoothing_range = 20;
517 *   unsigned int degree = 5;
518 *   unsigned int eig_cg_n_iterations = 20;
519 *   } smoother;
520 *  
521 *   struct
522 *   {
524 *   p_sequence = MGTransferGlobalCoarseningTools::
525 *   PolynomialCoarseningSequenceType::decrease_by_one;
526 *   bool perform_h_transfer = true;
527 *   } transfer;
528 *   };
529 *  
530 *  
531 *  
532 * @endcode
533 *
534 * This is the general parameter struct for the problem class. You will find
535 * this struct divided into several categories, including general runtime
536 * parameters, level limits, refine and coarsen fractions, as well as
537 * parameters for cell weighting. It also contains an instance of the above
538 * struct for multigrid parameters which will be passed to the multigrid
539 * algorithm.
540 *
541 * @code
542 *   struct Parameters
543 *   {
544 *   unsigned int n_cycles = 8;
545 *   double tolerance_factor = 1e-12;
546 *  
547 *   MultigridParameters mg_data;
548 *  
549 *   unsigned int min_h_level = 5;
550 *   unsigned int max_h_level = 12;
551 *   unsigned int min_p_degree = 2;
552 *   unsigned int max_p_degree = 6;
553 *   unsigned int max_p_level_difference = 1;
554 *  
555 *   double refine_fraction = 0.3;
556 *   double coarsen_fraction = 0.03;
557 *   double p_refine_fraction = 0.9;
558 *   double p_coarsen_fraction = 0.9;
559 *  
560 *   double weighting_factor = 1.;
561 *   double weighting_exponent = 1.;
562 *   };
563 *  
564 *  
565 *  
566 * @endcode
567 *
568 *
569 * <a name="step_75-MatrixfreeLaplaceoperator"></a>
570 * <h3>Matrix-free Laplace operator</h3>
571 *
572
573 *
574 * This is a matrix-free implementation of the Laplace operator that will
575 * basically take over the part of the `assemble_system()` function from other
576 * tutorials. The meaning of all member functions will be explained at their
577 * definition later.
578 *
579
580 *
581 * We will use the FEEvaluation class to evaluate the solution vector
582 * at the quadrature points and to perform the integration. In contrast to
583 * other tutorials, the template arguments `degree` is set to @f$-1@f$ and
584 * `number of quadrature in 1d` to @f$0@f$. In this case, FEEvaluation selects
585 * dynamically the correct polynomial degree and number of quadrature
586 * points. Here, we introduce an alias to FEEvaluation with the correct
587 * template parameters so that we do not have to worry about them later on.
588 *
589 * @code
590 *   template <int dim, typename number>
591 *   class LaplaceOperator : public Subscriptor
592 *   {
593 *   public:
594 *   using VectorType = LinearAlgebra::distributed::Vector<number>;
595 *  
596 *   using FECellIntegrator = FEEvaluation<dim, -1, 0, 1, number>;
597 *  
598 *   LaplaceOperator() = default;
599 *  
600 *   LaplaceOperator(const hp::MappingCollection<dim> &mapping,
601 *   const DoFHandler<dim> &dof_handler,
602 *   const hp::QCollection<dim> &quad,
603 *   const AffineConstraints<number> &constraints,
604 *   VectorType &system_rhs);
605 *  
606 *   void reinit(const hp::MappingCollection<dim> &mapping,
607 *   const DoFHandler<dim> &dof_handler,
608 *   const hp::QCollection<dim> &quad,
609 *   const AffineConstraints<number> &constraints,
610 *   VectorType &system_rhs);
611 *  
612 *   types::global_dof_index m() const;
613 *  
614 *   number el(unsigned int, unsigned int) const;
615 *  
616 *   void initialize_dof_vector(VectorType &vec) const;
617 *  
618 *   void vmult(VectorType &dst, const VectorType &src) const;
619 *  
620 *   void Tvmult(VectorType &dst, const VectorType &src) const;
621 *  
622 *   const TrilinosWrappers::SparseMatrix &get_system_matrix() const;
623 *  
624 *   void compute_inverse_diagonal(VectorType &diagonal) const;
625 *  
626 *   private:
627 *   void do_cell_integral_local(FECellIntegrator &integrator) const;
628 *  
629 *   void do_cell_integral_global(FECellIntegrator &integrator,
630 *   VectorType &dst,
631 *   const VectorType &src) const;
632 *  
633 *  
634 *   void do_cell_integral_range(
635 *   const MatrixFree<dim, number> &matrix_free,
636 *   VectorType &dst,
637 *   const VectorType &src,
638 *   const std::pair<unsigned int, unsigned int> &range) const;
639 *  
640 *   MatrixFree<dim, number> matrix_free;
641 *  
642 * @endcode
643 *
644 * To solve the equation system on the coarsest level with an AMG
645 * preconditioner, we need an actual system matrix on the coarsest level.
646 * For this purpose, we provide a mechanism that optionally computes a
647 * matrix from the matrix-free formulation, for which we introduce a
648 * dedicated SparseMatrix object. In the default case, this matrix stays
649 * empty. Once `get_system_matrix()` is called, this matrix is filled (lazy
650 * allocation). Since this is a `const` function, we need the "mutable"
651 * keyword here. We also need a the constraints object to build the matrix.
652 *
653 * @code
654 *   AffineConstraints<number> constraints;
655 *   mutable TrilinosWrappers::SparseMatrix system_matrix;
656 *   };
657 *  
658 *  
659 *  
660 * @endcode
661 *
662 * The following section contains functions to initialize and reinitialize
663 * the class. In particular, these functions initialize the internal
664 * MatrixFree instance. For sake of simplicity, we also compute the system
665 * right-hand-side vector.
666 *
667 * @code
668 *   template <int dim, typename number>
669 *   LaplaceOperator<dim, number>::LaplaceOperator(
670 *   const hp::MappingCollection<dim> &mapping,
671 *   const DoFHandler<dim> &dof_handler,
672 *   const hp::QCollection<dim> &quad,
673 *   const AffineConstraints<number> &constraints,
674 *   VectorType &system_rhs)
675 *   {
676 *   this->reinit(mapping, dof_handler, quad, constraints, system_rhs);
677 *   }
678 *  
679 *  
680 *  
681 *   template <int dim, typename number>
682 *   void LaplaceOperator<dim, number>::reinit(
683 *   const hp::MappingCollection<dim> &mapping,
684 *   const DoFHandler<dim> &dof_handler,
685 *   const hp::QCollection<dim> &quad,
686 *   const AffineConstraints<number> &constraints,
687 *   VectorType &system_rhs)
688 *   {
689 * @endcode
690 *
691 * Clear internal data structures (in the case that the operator is reused).
692 *
693 * @code
694 *   this->system_matrix.clear();
695 *  
696 * @endcode
697 *
698 * Copy the constraints, since they might be needed for computation of the
699 * system matrix later on.
700 *
701 * @code
702 *   this->constraints.copy_from(constraints);
703 *  
704 * @endcode
705 *
706 * Set up MatrixFree. At the quadrature points, we only need to evaluate
707 * the gradient of the solution and test with the gradient of the shape
708 * functions so that we only need to set the flag `update_gradients`.
709 *
710 * @code
713 *  
714 *   matrix_free.reinit(mapping, dof_handler, constraints, quad, data);
715 *  
716 * @endcode
717 *
718 * Compute the right-hand side vector. For this purpose, we set up a second
719 * MatrixFree instance that uses a modified AffineConstraints not containing
720 * the constraints due to Dirichlet-boundary conditions. This modified
721 * operator is applied to a vector with only the Dirichlet values set. The
722 * result is the negative right-hand-side vector.
723 *
724 * @code
725 *   {
726 *   AffineConstraints<number> constraints_without_dbc(
727 *   dof_handler.locally_owned_dofs(),
729 *  
731 *   constraints_without_dbc);
732 *   constraints_without_dbc.close();
733 *  
734 *   VectorType b, x;
735 *  
736 *   this->initialize_dof_vector(system_rhs);
737 *  
738 *   MatrixFree<dim, number> matrix_free;
739 *   matrix_free.reinit(
740 *   mapping, dof_handler, constraints_without_dbc, quad, data);
741 *  
742 *   matrix_free.initialize_dof_vector(b);
743 *   matrix_free.initialize_dof_vector(x);
744 *  
745 *   constraints.distribute(x);
746 *  
747 *   matrix_free.cell_loop(&LaplaceOperator::do_cell_integral_range,
748 *   this,
749 *   b,
750 *   x);
751 *  
752 *   constraints.set_zero(b);
753 *  
754 *   system_rhs -= b;
755 *   }
756 *   }
757 *  
758 *  
759 *  
760 * @endcode
761 *
762 * The following functions are implicitly needed by the multigrid algorithm,
763 * including the smoothers.
764 *
765
766 *
767 * Since we do not have a matrix, query the DoFHandler for the number of
768 * degrees of freedom.
769 *
770 * @code
771 *   template <int dim, typename number>
772 *   types::global_dof_index LaplaceOperator<dim, number>::m() const
773 *   {
774 *   return matrix_free.get_dof_handler().n_dofs();
775 *   }
776 *  
777 *  
778 *  
779 * @endcode
780 *
781 * Access a particular element in the matrix. This function is neither
782 * needed nor implemented, however, is required to compile the program.
783 *
784 * @code
785 *   template <int dim, typename number>
786 *   number LaplaceOperator<dim, number>::el(unsigned int, unsigned int) const
787 *   {
789 *   return 0;
790 *   }
791 *  
792 *  
793 *  
794 * @endcode
795 *
796 * Initialize the given vector. We simply delegate the task to the
797 * MatrixFree function with the same name.
798 *
799 * @code
800 *   template <int dim, typename number>
801 *   void
802 *   LaplaceOperator<dim, number>::initialize_dof_vector(VectorType &vec) const
803 *   {
804 *   matrix_free.initialize_dof_vector(vec);
805 *   }
806 *  
807 *  
808 *  
809 * @endcode
810 *
811 * Perform an operator evaluation by looping with the help of MatrixFree
812 * over all cells and evaluating the effect of the cell integrals (see also:
813 * `do_cell_integral_local()` and `do_cell_integral_global()`).
814 *
815 * @code
816 *   template <int dim, typename number>
817 *   void LaplaceOperator<dim, number>::vmult(VectorType &dst,
818 *   const VectorType &src) const
819 *   {
820 *   this->matrix_free.cell_loop(
821 *   &LaplaceOperator::do_cell_integral_range, this, dst, src, true);
822 *   }
823 *  
824 *  
825 *  
826 * @endcode
827 *
828 * Perform the transposed operator evaluation. Since we are considering
829 * symmetric "matrices", this function can simply delegate it task to vmult().
830 *
831 * @code
832 *   template <int dim, typename number>
833 *   void LaplaceOperator<dim, number>::Tvmult(VectorType &dst,
834 *   const VectorType &src) const
835 *   {
836 *   this->vmult(dst, src);
837 *   }
838 *  
839 *  
840 *  
841 * @endcode
842 *
843 * Since we do not have a system matrix, we cannot loop over the the
844 * diagonal entries of the matrix. Instead, we compute the diagonal by
845 * performing a sequence of operator evaluations to unit basis vectors.
846 * For this purpose, an optimized function from the MatrixFreeTools
847 * namespace is used. The inversion is performed manually afterwards.
848 *
849 * @code
850 *   template <int dim, typename number>
851 *   void LaplaceOperator<dim, number>::compute_inverse_diagonal(
852 *   VectorType &diagonal) const
853 *   {
854 *   this->matrix_free.initialize_dof_vector(diagonal);
855 *   MatrixFreeTools::compute_diagonal(matrix_free,
856 *   diagonal,
857 *   &LaplaceOperator::do_cell_integral_local,
858 *   this);
859 *  
860 *   for (auto &i : diagonal)
861 *   i = (std::abs(i) > 1.0e-10) ? (1.0 / i) : 1.0;
862 *   }
863 *  
864 *  
865 *  
866 * @endcode
867 *
868 * In the matrix-free context, no system matrix is set up during
869 * initialization of this class. As a consequence, it has to be computed
870 * here if it should be requested. Since the matrix is only computed in
871 * this tutorial for linear elements (on the coarse grid), this is
872 * acceptable.
873 * The matrix entries are obtained via sequence of operator evaluations.
874 * For this purpose, the optimized function MatrixFreeTools::compute_matrix()
875 * is used. The matrix will only be computed if it has not been set up yet
876 * (lazy allocation).
877 *
878 * @code
879 *   template <int dim, typename number>
881 *   LaplaceOperator<dim, number>::get_system_matrix() const
882 *   {
883 *   if (system_matrix.m() == 0 && system_matrix.n() == 0)
884 *   {
885 *   const auto &dof_handler = this->matrix_free.get_dof_handler();
886 *  
888 *   dof_handler.locally_owned_dofs(),
889 *   dof_handler.get_triangulation().get_communicator());
890 *  
891 *   DoFTools::make_sparsity_pattern(dof_handler, dsp, this->constraints);
892 *  
893 *   dsp.compress();
894 *   system_matrix.reinit(dsp);
895 *  
897 *   matrix_free,
898 *   constraints,
899 *   system_matrix,
900 *   &LaplaceOperator::do_cell_integral_local,
901 *   this);
902 *   }
903 *  
904 *   return this->system_matrix;
905 *   }
906 *  
907 *  
908 *  
909 * @endcode
910 *
911 * Perform cell integral on a cell batch without gathering and scattering
912 * the values. This function is needed for the MatrixFreeTools functions
913 * since these functions operate directly on the buffers of FEEvaluation.
914 *
915 * @code
916 *   template <int dim, typename number>
917 *   void LaplaceOperator<dim, number>::do_cell_integral_local(
918 *   FECellIntegrator &integrator) const
919 *   {
920 *   integrator.evaluate(EvaluationFlags::gradients);
921 *  
922 *   for (const unsigned int q : integrator.quadrature_point_indices())
923 *   integrator.submit_gradient(integrator.get_gradient(q), q);
924 *  
925 *   integrator.integrate(EvaluationFlags::gradients);
926 *   }
927 *  
928 *  
929 *  
930 * @endcode
931 *
932 * Same as above but with access to the global vectors.
933 *
934 * @code
935 *   template <int dim, typename number>
936 *   void LaplaceOperator<dim, number>::do_cell_integral_global(
937 *   FECellIntegrator &integrator,
938 *   VectorType &dst,
939 *   const VectorType &src) const
940 *   {
941 *   integrator.gather_evaluate(src, EvaluationFlags::gradients);
942 *  
943 *   for (const unsigned int q : integrator.quadrature_point_indices())
944 *   integrator.submit_gradient(integrator.get_gradient(q), q);
945 *  
946 *   integrator.integrate_scatter(EvaluationFlags::gradients, dst);
947 *   }
948 *  
949 *  
950 *  
951 * @endcode
952 *
953 * This function loops over all cell batches within a cell-batch range and
954 * calls the above function.
955 *
956 * @code
957 *   template <int dim, typename number>
958 *   void LaplaceOperator<dim, number>::do_cell_integral_range(
959 *   const MatrixFree<dim, number> &matrix_free,
960 *   VectorType &dst,
961 *   const VectorType &src,
962 *   const std::pair<unsigned int, unsigned int> &range) const
963 *   {
964 *   FECellIntegrator integrator(matrix_free, range);
965 *  
966 *   for (unsigned cell = range.first; cell < range.second; ++cell)
967 *   {
968 *   integrator.reinit(cell);
969 *  
970 *   do_cell_integral_global(integrator, dst, src);
971 *   }
972 *   }
973 *  
974 *  
975 *  
976 * @endcode
977 *
978 *
979 * <a name="step_75-Solverandpreconditioner"></a>
980 * <h3>Solver and preconditioner</h3>
981 *
982
983 *
984 *
985 * <a name="step_75-Conjugategradientsolverwithmultigridpreconditioner"></a>
986 * <h4>Conjugate-gradient solver with multigrid preconditioner</h4>
987 *
988
989 *
990 * This function solves the equation system with a sequence of provided
991 * multigrid objects. It is meant to be treated as general as possible, hence
992 * the multitude of template parameters.
993 *
994 * @code
995 *   template <typename VectorType,
996 *   int dim,
997 *   typename SystemMatrixType,
998 *   typename LevelMatrixType,
999 *   typename MGTransferType>
1000 *   static void
1001 *   mg_solve(SolverControl &solver_control,
1002 *   VectorType &dst,
1003 *   const VectorType &src,
1004 *   const MultigridParameters &mg_data,
1005 *   const DoFHandler<dim> &dof,
1006 *   const SystemMatrixType &fine_matrix,
1007 *   const MGLevelObject<std::unique_ptr<LevelMatrixType>> &mg_matrices,
1008 *   const MGTransferType &mg_transfer)
1009 *   {
1010 *   AssertThrow(mg_data.coarse_solver.type == "cg_with_amg",
1011 *   ExcNotImplemented());
1012 *   AssertThrow(mg_data.smoother.type == "chebyshev", ExcNotImplemented());
1013 *  
1014 *   const unsigned int min_level = mg_matrices.min_level();
1015 *   const unsigned int max_level = mg_matrices.max_level();
1016 *  
1017 *   using SmootherPreconditionerType = DiagonalMatrix<VectorType>;
1018 *   using SmootherType = PreconditionChebyshev<LevelMatrixType,
1019 *   VectorType,
1020 *   SmootherPreconditionerType>;
1021 *   using PreconditionerType = PreconditionMG<dim, VectorType, MGTransferType>;
1022 *  
1023 * @endcode
1024 *
1025 * We initialize level operators and Chebyshev smoothers here.
1026 *
1027 * @code
1028 *   mg::Matrix<VectorType> mg_matrix(mg_matrices);
1029 *  
1031 *   min_level, max_level);
1032 *  
1033 *   for (unsigned int level = min_level; level <= max_level; ++level)
1034 *   {
1035 *   smoother_data[level].preconditioner =
1036 *   std::make_shared<SmootherPreconditionerType>();
1037 *   mg_matrices[level]->compute_inverse_diagonal(
1038 *   smoother_data[level].preconditioner->get_vector());
1039 *   smoother_data[level].smoothing_range = mg_data.smoother.smoothing_range;
1040 *   smoother_data[level].degree = mg_data.smoother.degree;
1041 *   smoother_data[level].eig_cg_n_iterations =
1042 *   mg_data.smoother.eig_cg_n_iterations;
1043 *   }
1044 *  
1046 *   mg_smoother;
1047 *   mg_smoother.initialize(mg_matrices, smoother_data);
1048 *  
1049 * @endcode
1050 *
1051 * Next, we initialize the coarse-grid solver. We use conjugate-gradient
1052 * method with AMG as preconditioner.
1053 *
1054 * @code
1055 *   ReductionControl coarse_grid_solver_control(mg_data.coarse_solver.maxiter,
1056 *   mg_data.coarse_solver.abstol,
1057 *   mg_data.coarse_solver.reltol,
1058 *   false,
1059 *   false);
1060 *   SolverCG<VectorType> coarse_grid_solver(coarse_grid_solver_control);
1061 *  
1062 *   std::unique_ptr<MGCoarseGridBase<VectorType>> mg_coarse;
1063 *  
1064 *   TrilinosWrappers::PreconditionAMG precondition_amg;
1066 *   amg_data.smoother_sweeps = mg_data.coarse_solver.smoother_sweeps;
1067 *   amg_data.n_cycles = mg_data.coarse_solver.n_cycles;
1068 *   amg_data.smoother_type = mg_data.coarse_solver.smoother_type.c_str();
1069 *  
1070 *   precondition_amg.initialize(mg_matrices[min_level]->get_system_matrix(),
1071 *   amg_data);
1072 *  
1073 *   mg_coarse =
1074 *   std::make_unique<MGCoarseGridIterativeSolver<VectorType,
1076 *   LevelMatrixType,
1077 *   decltype(precondition_amg)>>(
1078 *   coarse_grid_solver, *mg_matrices[min_level], precondition_amg);
1079 *  
1080 * @endcode
1081 *
1082 * Finally, we create the Multigrid object, convert it to a preconditioner,
1083 * and use it inside of a conjugate-gradient solver to solve the linear
1084 * system of equations.
1085 *
1086 * @code
1088 *   mg_matrix, *mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1089 *  
1090 *   PreconditionerType preconditioner(dof, mg, mg_transfer);
1091 *  
1092 *   SolverCG<VectorType>(solver_control)
1093 *   .solve(fine_matrix, dst, src, preconditioner);
1094 *   }
1095 *  
1096 *  
1097 *  
1098 * @endcode
1099 *
1100 *
1101 * <a name="step_75-Hybridpolynomialgeometricglobalcoarseningmultigridpreconditioner"></a>
1102 * <h4>Hybrid polynomial/geometric-global-coarsening multigrid preconditioner</h4>
1103 *
1104
1105 *
1106 * The above function deals with the actual solution for a given sequence of
1107 * multigrid objects. This functions creates the actual multigrid levels, in
1108 * particular the operators, and the transfer operator as a
1110 *
1111 * @code
1112 *   template <typename VectorType, typename OperatorType, int dim>
1113 *   void solve_with_gmg(SolverControl &solver_control,
1114 *   const OperatorType &system_matrix,
1115 *   VectorType &dst,
1116 *   const VectorType &src,
1117 *   const MultigridParameters &mg_data,
1118 *   const hp::MappingCollection<dim> mapping_collection,
1119 *   const DoFHandler<dim> &dof_handler,
1120 *   const hp::QCollection<dim> &quadrature_collection)
1121 *   {
1122 * @endcode
1123 *
1124 * Create a DoFHandler and operator for each multigrid level,
1125 * as well as, create transfer operators. To be able to
1126 * set up the operators, we need a set of DoFHandler that we create
1127 * via global coarsening of p or h. For latter, we need also a sequence
1128 * of Triangulation objects that are obtained by
1130 *
1131
1132 *
1133 * In case no h-transfer is requested, we provide an empty deleter for the
1134 * `emplace_back()` function, since the Triangulation of our DoFHandler is
1135 * an external field and its destructor is called somewhere else.
1136 *
1137 * @code
1138 *   MGLevelObject<DoFHandler<dim>> dof_handlers;
1141 *  
1142 *   std::vector<std::shared_ptr<const Triangulation<dim>>>
1143 *   coarse_grid_triangulations;
1144 *   if (mg_data.transfer.perform_h_transfer)
1145 *   coarse_grid_triangulations =
1147 *   dof_handler.get_triangulation());
1148 *   else
1149 *   coarse_grid_triangulations.emplace_back(
1150 *   &(dof_handler.get_triangulation()), [](auto *) {});
1151 *  
1152 * @endcode
1153 *
1154 * Determine the total number of levels for the multigrid operation and
1155 * allocate sufficient memory for all levels.
1156 *
1157 * @code
1158 *   const unsigned int n_h_levels = coarse_grid_triangulations.size() - 1;
1159 *  
1160 *   const auto get_max_active_fe_degree = [&](const auto &dof_handler) {
1161 *   unsigned int max = 0;
1162 *  
1163 *   for (auto &cell : dof_handler.active_cell_iterators())
1164 *   if (cell->is_locally_owned())
1165 *   max =
1166 *   std::max(max, dof_handler.get_fe(cell->active_fe_index()).degree);
1167 *  
1168 *   return Utilities::MPI::max(max, MPI_COMM_WORLD);
1169 *   };
1170 *  
1171 *   const unsigned int n_p_levels =
1173 *   get_max_active_fe_degree(dof_handler), mg_data.transfer.p_sequence)
1174 *   .size();
1175 *  
1176 *   std::map<unsigned int, unsigned int> fe_index_for_degree;
1177 *   for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
1178 *   {
1179 *   const unsigned int degree = dof_handler.get_fe(i).degree;
1180 *   Assert(fe_index_for_degree.find(degree) == fe_index_for_degree.end(),
1181 *   ExcMessage("FECollection does not contain unique degrees."));
1182 *   fe_index_for_degree[degree] = i;
1183 *   }
1184 *  
1185 *   unsigned int minlevel = 0;
1186 *   unsigned int maxlevel = n_h_levels + n_p_levels - 1;
1187 *  
1188 *   dof_handlers.resize(minlevel, maxlevel);
1189 *   operators.resize(minlevel, maxlevel);
1190 *   transfers.resize(minlevel, maxlevel);
1191 *  
1192 * @endcode
1193 *
1194 * Loop from the minimum (coarsest) to the maximum (finest) level and set up
1195 * DoFHandler accordingly. We start with the h-levels, where we distribute
1196 * on increasingly finer meshes linear elements.
1197 *
1198 * @code
1199 *   for (unsigned int l = 0; l < n_h_levels; ++l)
1200 *   {
1201 *   dof_handlers[l].reinit(*coarse_grid_triangulations[l]);
1202 *   dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
1203 *   }
1204 *  
1205 * @endcode
1206 *
1207 * After we reached the finest mesh, we will adjust the polynomial degrees
1208 * on each level. We reverse iterate over our data structure and start at
1209 * the finest mesh that contains all information about the active FE
1210 * indices. We then lower the polynomial degree of each cell level by level.
1211 *
1212 * @code
1213 *   for (unsigned int i = 0, l = maxlevel; i < n_p_levels; ++i, --l)
1214 *   {
1215 *   dof_handlers[l].reinit(dof_handler.get_triangulation());
1216 *  
1217 *   if (l == maxlevel) // finest level
1218 *   {
1219 *   auto &dof_handler_mg = dof_handlers[l];
1220 *  
1221 *   auto cell_other = dof_handler.begin_active();
1222 *   for (auto &cell : dof_handler_mg.active_cell_iterators())
1223 *   {
1224 *   if (cell->is_locally_owned())
1225 *   cell->set_active_fe_index(cell_other->active_fe_index());
1226 *   ++cell_other;
1227 *   }
1228 *   }
1229 *   else // coarse level
1230 *   {
1231 *   auto &dof_handler_fine = dof_handlers[l + 1];
1232 *   auto &dof_handler_coarse = dof_handlers[l + 0];
1233 *  
1234 *   auto cell_other = dof_handler_fine.begin_active();
1235 *   for (auto &cell : dof_handler_coarse.active_cell_iterators())
1236 *   {
1237 *   if (cell->is_locally_owned())
1238 *   {
1239 *   const unsigned int next_degree =
1242 *   cell_other->get_fe().degree,
1243 *   mg_data.transfer.p_sequence);
1244 *   Assert(fe_index_for_degree.find(next_degree) !=
1245 *   fe_index_for_degree.end(),
1246 *   ExcMessage("Next polynomial degree in sequence "
1247 *   "does not exist in FECollection."));
1248 *  
1249 *   cell->set_active_fe_index(fe_index_for_degree[next_degree]);
1250 *   }
1251 *   ++cell_other;
1252 *   }
1253 *   }
1254 *  
1255 *   dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
1256 *   }
1257 *  
1258 * @endcode
1259 *
1260 * Next, we will create all data structures additionally needed on each
1261 * multigrid level. This involves determining constraints with homogeneous
1262 * Dirichlet boundary conditions, and building the operator just like on the
1263 * active level.
1264 *
1265 * @code
1267 *   constraints(minlevel, maxlevel);
1268 *  
1269 *   for (unsigned int level = minlevel; level <= maxlevel; ++level)
1270 *   {
1271 *   const auto &dof_handler = dof_handlers[level];
1272 *   auto &constraint = constraints[level];
1273 *  
1274 *   constraint.reinit(dof_handler.locally_owned_dofs(),
1275 *   DoFTools::extract_locally_relevant_dofs(dof_handler));
1276 *  
1277 *   DoFTools::make_hanging_node_constraints(dof_handler, constraint);
1278 *   VectorTools::interpolate_boundary_values(mapping_collection,
1279 *   dof_handler,
1280 *   0,
1282 *   constraint);
1283 *   constraint.close();
1284 *  
1285 *   VectorType dummy;
1286 *  
1287 *   operators[level] = std::make_unique<OperatorType>(mapping_collection,
1288 *   dof_handler,
1289 *   quadrature_collection,
1290 *   constraint,
1291 *   dummy);
1292 *   }
1293 *  
1294 * @endcode
1295 *
1296 * Set up intergrid operators and collect transfer operators within a single
1297 * operator as needed by the Multigrid solver class.
1298 *
1299 * @code
1300 *   for (unsigned int level = minlevel; level < maxlevel; ++level)
1301 *   transfers[level + 1].reinit(dof_handlers[level + 1],
1302 *   dof_handlers[level],
1303 *   constraints[level + 1],
1304 *   constraints[level]);
1305 *  
1307 *   transfers, [&](const auto l, auto &vec) {
1308 *   operators[l]->initialize_dof_vector(vec);
1309 *   });
1310 *  
1311 * @endcode
1312 *
1313 * Finally, proceed to solve the problem with multigrid.
1314 *
1315 * @code
1316 *   mg_solve(solver_control,
1317 *   dst,
1318 *   src,
1319 *   mg_data,
1320 *   dof_handler,
1321 *   system_matrix,
1322 *   operators,
1323 *   transfer);
1324 *   }
1325 *  
1326 *  
1327 *  
1328 * @endcode
1329 *
1330 *
1331 * <a name="step_75-ThecodeLaplaceProblemcodeclasstemplate"></a>
1332 * <h3>The <code>LaplaceProblem</code> class template</h3>
1333 *
1334
1335 *
1336 * Now we will finally declare the main class of this program, which solves
1337 * the Laplace equation on subsequently refined function spaces. Its structure
1338 * will look familiar as it is similar to the main classes of @ref step_27 "step-27" and
1339 * @ref step_40 "step-40". There are basically just two additions:
1340 * - The SparseMatrix object that would hold the system matrix has been
1341 * replaced by an object of the LaplaceOperator class for the MatrixFree
1342 * formulation.
1343 * - An object of parallel::CellWeights, which will help us with load
1344 * balancing, has been added.
1345 *
1346 * @code
1347 *   template <int dim>
1348 *   class LaplaceProblem
1349 *   {
1350 *   public:
1351 *   LaplaceProblem(const Parameters &parameters);
1352 *  
1353 *   void run();
1354 *  
1355 *   private:
1356 *   void initialize_grid();
1357 *   void setup_system();
1358 *   void print_diagnostics();
1359 *   void solve_system();
1360 *   void compute_indicators();
1361 *   void adapt_resolution();
1362 *   void output_results(const unsigned int cycle);
1363 *  
1364 *   MPI_Comm mpi_communicator;
1365 *  
1366 *   const Parameters prm;
1367 *  
1369 *   DoFHandler<dim> dof_handler;
1370 *  
1371 *   hp::MappingCollection<dim> mapping_collection;
1372 *   hp::FECollection<dim> fe_collection;
1373 *   hp::QCollection<dim> quadrature_collection;
1374 *   hp::QCollection<dim - 1> face_quadrature_collection;
1375 *  
1376 *   IndexSet locally_owned_dofs;
1377 *   IndexSet locally_relevant_dofs;
1378 *  
1379 *   AffineConstraints<double> constraints;
1380 *  
1381 *   LaplaceOperator<dim, double> laplace_operator;
1382 *   LinearAlgebra::distributed::Vector<double> locally_relevant_solution;
1384 *  
1385 *   std::unique_ptr<FESeries::Legendre<dim>> legendre;
1386 *   std::unique_ptr<parallel::CellWeights<dim>> cell_weights;
1387 *  
1388 *   Vector<float> estimated_error_per_cell;
1389 *   Vector<float> hp_decision_indicators;
1390 *  
1391 *   ConditionalOStream pcout;
1392 *   TimerOutput computing_timer;
1393 *   };
1394 *  
1395 *  
1396 *  
1397 * @endcode
1398 *
1399 *
1400 * <a name="step_75-ThecodeLaplaceProblemcodeclassimplementation"></a>
1401 * <h3>The <code>LaplaceProblem</code> class implementation</h3>
1402 *
1403
1404 *
1405 *
1406 * <a name="step_75-Constructor"></a>
1407 * <h4>Constructor</h4>
1408 *
1409
1410 *
1411 * The constructor starts with an initializer list that looks similar to the
1412 * one of @ref step_40 "step-40". We again prepare the ConditionalOStream object to allow
1413 * only the first process to output anything over the console, and initialize
1414 * the computing timer properly.
1415 *
1416 * @code
1417 *   template <int dim>
1418 *   LaplaceProblem<dim>::LaplaceProblem(const Parameters &parameters)
1419 *   : mpi_communicator(MPI_COMM_WORLD)
1420 *   , prm(parameters)
1421 *   , triangulation(mpi_communicator)
1422 *   , dof_handler(triangulation)
1423 *   , pcout(std::cout,
1424 *   (Utilities::MPI::this_mpi_process(mpi_communicator) == 0))
1425 *   , computing_timer(mpi_communicator,
1426 *   pcout,
1427 *   TimerOutput::never,
1429 *   {
1430 *   Assert(prm.min_h_level <= prm.max_h_level,
1431 *   ExcMessage(
1432 *   "Triangulation level limits have been incorrectly set up."));
1433 *   Assert(prm.min_p_degree <= prm.max_p_degree,
1434 *   ExcMessage("FECollection degrees have been incorrectly set up."));
1435 *  
1436 * @endcode
1437 *
1438 * We need to prepare the data structures for the hp-functionality in the
1439 * actual body of the constructor, and create corresponding objects for
1440 * every degree in the specified range from the parameter struct. As we are
1441 * only dealing with non-distorted rectangular cells, a linear mapping
1442 * object is sufficient in this context.
1443 *
1444
1445 *
1446 * In the Parameters struct, we provide ranges for levels on which the
1447 * function space is operating with a reasonable resolution. The multigrid
1448 * algorithm requires linear elements on the coarsest possible level. So we
1449 * start with the lowest polynomial degree and fill the collection with
1450 * consecutively higher degrees until the user-specified maximum is
1451 * reached.
1452 *
1453 * @code
1454 *   mapping_collection.push_back(MappingQ1<dim>());
1455 *  
1456 *   for (unsigned int degree = 1; degree <= prm.max_p_degree; ++degree)
1457 *   {
1458 *   fe_collection.push_back(FE_Q<dim>(degree));
1459 *   quadrature_collection.push_back(QGauss<dim>(degree + 1));
1460 *   face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
1461 *   }
1462 *  
1463 * @endcode
1464 *
1465 * As our FECollection contains more finite elements than we want to use for
1466 * the finite element approximation of our solution, we would like to limit
1467 * the range on which active FE indices can operate on. For this, the
1468 * FECollection class allows to register a hierarchy that determines the
1469 * succeeding and preceding finite element in case of of p-refinement and
1470 * p-coarsening, respectively. All functions in the hp::Refinement namespace
1471 * consult this hierarchy to determine future FE indices. We will register
1472 * such a hierarchy that only works on finite elements with polynomial
1473 * degrees in the proposed range <code>[min_p_degree, max_p_degree]</code>.
1474 *
1475 * @code
1476 *   const unsigned int min_fe_index = prm.min_p_degree - 1;
1477 *   fe_collection.set_hierarchy(
1478 *   /*next_index=*/
1479 *   [](const typename hp::FECollection<dim> &fe_collection,
1480 *   const unsigned int fe_index) -> unsigned int {
1481 *   return ((fe_index + 1) < fe_collection.size()) ? fe_index + 1 :
1482 *   fe_index;
1483 *   },
1484 *   /*previous_index=*/
1485 *   [min_fe_index](const typename hp::FECollection<dim> &,
1486 *   const unsigned int fe_index) -> unsigned int {
1487 *   Assert(fe_index >= min_fe_index,
1488 *   ExcMessage("Finite element is not part of hierarchy!"));
1489 *   return (fe_index > min_fe_index) ? fe_index - 1 : fe_index;
1490 *   });
1491 *  
1492 * @endcode
1493 *
1494 * We initialize the FESeries::Legendre object in the default configuration
1495 * for smoothness estimation.
1496 *
1497 * @code
1498 *   legendre = std::make_unique<FESeries::Legendre<dim>>(
1500 *  
1501 * @endcode
1502 *
1503 * The next part is going to be tricky. During execution of refinement, a
1504 * few hp-algorithms need to interfere with the actual refinement process on
1505 * the Triangulation object. We do this by connecting several functions to
1506 * Triangulation::Signals: signals will be called at different stages during
1507 * the actual refinement process and trigger all connected functions. We
1508 * require this functionality for load balancing and to limit the polynomial
1509 * degrees of neighboring cells.
1510 *
1511
1512 *
1513 * For the former, we would like to assign a weight to every cell that is
1514 * proportional to the number of degrees of freedom of its future finite
1515 * element. The library offers a class parallel::CellWeights that allows to
1516 * easily attach individual weights at the right place during the refinement
1517 * process, i.e., after all refine and coarsen flags have been set correctly
1518 * for hp-adaptation and right before repartitioning for load balancing is
1519 * about to happen. Functions can be registered that will attach weights in
1520 * the form that @f$a (n_\text{dofs})^b@f$ with a provided pair of parameters
1521 * @f$(a,b)@f$. We register such a function in the following.
1522 *
1523
1524 *
1525 * For load balancing, efficient solvers like the one we use should scale
1526 * linearly with the number of degrees of freedom owned. We set the
1527 * parameters for cell weighting correspondingly: A weighting factor of @f$1@f$
1528 * and an exponent of @f$1@f$ (see the definitions of the `weighting_factor` and
1529 * `weighting_exponent` above).
1530 *
1531 * @code
1532 *   cell_weights = std::make_unique<parallel::CellWeights<dim>>(
1533 *   dof_handler,
1535 *   {prm.weighting_factor, prm.weighting_exponent}));
1536 *  
1537 * @endcode
1538 *
1539 * In h-adaptive applications, we ensure a 2:1 mesh balance by limiting the
1540 * difference of refinement levels of neighboring cells to one. With the
1541 * second call in the following code snippet, we will ensure the same for
1542 * p-levels on neighboring cells: levels of future finite elements are not
1543 * allowed to differ by more than a specified difference. The function
1544 * hp::Refinement::limit_p_level_difference takes care of this, but needs to
1545 * be connected to a very specific signal in the parallel context. The issue
1546 * is that we need to know how the mesh will be actually refined to set
1547 * future FE indices accordingly. As we ask the p4est oracle to perform
1548 * refinement, we need to ensure that the Triangulation has been updated
1549 * with the adaptation flags of the oracle first. An instantiation of
1551 * that for the duration of its life. Thus, we will create an object of this
1552 * class right before limiting the p-level difference, and connect the
1553 * corresponding lambda function to the signal
1554 * Triangulation::Signals::post_p4est_refinement, which will be triggered
1555 * after the oracle got refined, but before the Triangulation is refined.
1556 * Furthermore, we specify that this function will be connected to the front
1557 * of the signal, to ensure that the modification is performed before any
1558 * other function connected to the same signal.
1559 *
1560 * @code
1561 *   triangulation.signals.post_p4est_refinement.connect(
1562 *   [&, min_fe_index]() {
1564 *   refine_modifier(triangulation);
1566 *   prm.max_p_level_difference,
1567 *   /*contains=*/min_fe_index);
1568 *   },
1569 *   boost::signals2::at_front);
1570 *   }
1571 *  
1572 *  
1573 *  
1574 * @endcode
1575 *
1576 *
1577 * <a name="step_75-LaplaceProbleminitialize_grid"></a>
1578 * <h4>LaplaceProblem::initialize_grid</h4>
1579 *
1580
1581 *
1582 * For a L-shaped domain, we could use the function GridGenerator::hyper_L()
1583 * as demonstrated in @ref step_50 "step-50". However in the 2d case, that particular
1584 * function removes the first quadrant, while we need the fourth quadrant
1585 * removed in our scenario. Thus, we will use a different function
1586 * GridGenerator::subdivided_hyper_L() which gives us more options to create
1587 * the mesh. Furthermore, we formulate that function in a way that it also
1588 * generates a 3d mesh: the 2d L-shaped domain will basically elongated by 1
1589 * in the positive z-direction.
1590 *
1591
1592 *
1593 * We first pretend to build a GridGenerator::subdivided_hyper_rectangle().
1594 * The parameters that we need to provide are Point objects for the lower left
1595 * and top right corners, as well as the number of repetitions that the base
1596 * mesh will have in each direction. We provide them for the first two
1597 * dimensions and treat the higher third dimension separately.
1598 *
1599
1600 *
1601 * To create a L-shaped domain, we need to remove the excess cells. For this,
1602 * we specify the <code>cells_to_remove</code> accordingly. We would like to
1603 * remove one cell in every cell from the negative direction, but remove one
1604 * from the positive x-direction.
1605 *
1606
1607 *
1608 * On the coarse grid, we set the initial active FE indices and distribute the
1609 * degrees of freedom once. We do that in order to assign the hp::FECollection
1610 * to the DoFHandler, so that all cells know how many DoFs they are going to
1611 * have. This step is mandatory for the weighted load balancing algorithm,
1612 * which will be called implicitly in
1613 * parallel::distributed::Triangulation::refine_global().
1614 *
1615 * @code
1616 *   template <int dim>
1617 *   void LaplaceProblem<dim>::initialize_grid()
1618 *   {
1619 *   TimerOutput::Scope t(computing_timer, "initialize grid");
1620 *  
1621 *   std::vector<unsigned int> repetitions(dim);
1622 *   Point<dim> bottom_left, top_right;
1623 *   for (unsigned int d = 0; d < dim; ++d)
1624 *   if (d < 2)
1625 *   {
1626 *   repetitions[d] = 2;
1627 *   bottom_left[d] = -1.;
1628 *   top_right[d] = 1.;
1629 *   }
1630 *   else
1631 *   {
1632 *   repetitions[d] = 1;
1633 *   bottom_left[d] = 0.;
1634 *   top_right[d] = 1.;
1635 *   }
1636 *  
1637 *   std::vector<int> cells_to_remove(dim, 1);
1638 *   cells_to_remove[0] = -1;
1639 *  
1641 *   triangulation, repetitions, bottom_left, top_right, cells_to_remove);
1642 *  
1643 *   const unsigned int min_fe_index = prm.min_p_degree - 1;
1644 *   for (const auto &cell : dof_handler.active_cell_iterators())
1645 *   if (cell->is_locally_owned())
1646 *   cell->set_active_fe_index(min_fe_index);
1647 *  
1648 *   dof_handler.distribute_dofs(fe_collection);
1649 *  
1650 *   triangulation.refine_global(prm.min_h_level);
1651 *   }
1652 *  
1653 *  
1654 *  
1655 * @endcode
1656 *
1657 *
1658 * <a name="step_75-LaplaceProblemsetup_system"></a>
1659 * <h4>LaplaceProblem::setup_system</h4>
1660 *
1661
1662 *
1663 * This function looks exactly the same to the one of @ref step_40 "step-40", but you will
1664 * notice the absence of the system matrix as well as the scaffold that
1665 * surrounds it. Instead, we will initialize the MatrixFree formulation of the
1666 * <code>laplace_operator</code> here. For boundary conditions, we will use
1667 * the Solution class introduced earlier in this tutorial.
1668 *
1669 * @code
1670 *   template <int dim>
1671 *   void LaplaceProblem<dim>::setup_system()
1672 *   {
1673 *   TimerOutput::Scope t(computing_timer, "setup system");
1674 *  
1675 *   dof_handler.distribute_dofs(fe_collection);
1676 *  
1677 *   locally_owned_dofs = dof_handler.locally_owned_dofs();
1678 *   locally_relevant_dofs =
1680 *  
1681 *   locally_relevant_solution.reinit(locally_owned_dofs,
1682 *   locally_relevant_dofs,
1683 *   mpi_communicator);
1684 *   system_rhs.reinit(locally_owned_dofs, mpi_communicator);
1685 *  
1686 *   constraints.clear();
1687 *   constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
1688 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
1690 *   mapping_collection, dof_handler, 0, Solution<dim>(), constraints);
1691 *   constraints.close();
1692 *  
1693 *   laplace_operator.reinit(mapping_collection,
1694 *   dof_handler,
1695 *   quadrature_collection,
1696 *   constraints,
1697 *   system_rhs);
1698 *   }
1699 *  
1700 *  
1701 *  
1702 * @endcode
1703 *
1704 *
1705 * <a name="step_75-LaplaceProblemprint_diagnostics"></a>
1706 * <h4>LaplaceProblem::print_diagnostics</h4>
1707 *
1708
1709 *
1710 * This is a function that prints additional diagnostics about the equation
1711 * system and its partitioning. In addition to the usual global number of
1712 * active cells and degrees of freedom, we also output their local
1713 * equivalents. For a regulated output, we will communicate the local
1714 * quantities with a Utilities::MPI::gather operation to the first process
1715 * which will then output all information. Output of local quantities is
1716 * limited to the first 8 processes to avoid cluttering the terminal.
1717 *
1718
1719 *
1720 * Furthermore, we would like to print the frequencies of the polynomial
1721 * degrees in the numerical discretization. Since this information is only
1722 * stored locally, we will count the finite elements on locally owned cells
1723 * and later communicate them via Utilities::MPI::sum.
1724 *
1725 * @code
1726 *   template <int dim>
1727 *   void LaplaceProblem<dim>::print_diagnostics()
1728 *   {
1729 *   const unsigned int first_n_processes =
1731 *   Utilities::MPI::n_mpi_processes(mpi_communicator));
1732 *   const bool output_cropped =
1733 *   first_n_processes < Utilities::MPI::n_mpi_processes(mpi_communicator);
1734 *  
1735 *   {
1736 *   pcout << " Number of active cells: "
1737 *   << triangulation.n_global_active_cells() << std::endl
1738 *   << " by partition: ";
1739 *  
1740 *   std::vector<unsigned int> n_active_cells_per_subdomain =
1741 *   Utilities::MPI::gather(mpi_communicator,
1743 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1744 *   pcout << ' ' << n_active_cells_per_subdomain[i];
1745 *   if (output_cropped)
1746 *   pcout << " ...";
1747 *   pcout << std::endl;
1748 *   }
1749 *  
1750 *   {
1751 *   pcout << " Number of degrees of freedom: " << dof_handler.n_dofs()
1752 *   << std::endl
1753 *   << " by partition: ";
1754 *  
1755 *   std::vector<types::global_dof_index> n_dofs_per_subdomain =
1756 *   Utilities::MPI::gather(mpi_communicator,
1757 *   dof_handler.n_locally_owned_dofs());
1758 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1759 *   pcout << ' ' << n_dofs_per_subdomain[i];
1760 *   if (output_cropped)
1761 *   pcout << " ...";
1762 *   pcout << std::endl;
1763 *   }
1764 *  
1765 *   {
1766 *   std::vector<types::global_dof_index> n_constraints_per_subdomain =
1767 *   Utilities::MPI::gather(mpi_communicator, constraints.n_constraints());
1768 *  
1769 *   pcout << " Number of constraints: "
1770 *   << std::accumulate(n_constraints_per_subdomain.begin(),
1771 *   n_constraints_per_subdomain.end(),
1772 *   0)
1773 *   << std::endl
1774 *   << " by partition: ";
1775 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1776 *   pcout << ' ' << n_constraints_per_subdomain[i];
1777 *   if (output_cropped)
1778 *   pcout << " ...";
1779 *   pcout << std::endl;
1780 *   }
1781 *  
1782 *   {
1783 *   std::vector<unsigned int> n_fe_indices(fe_collection.size(), 0);
1784 *   for (const auto &cell : dof_handler.active_cell_iterators())
1785 *   if (cell->is_locally_owned())
1786 *   n_fe_indices[cell->active_fe_index()]++;
1787 *  
1788 *   Utilities::MPI::sum(n_fe_indices, mpi_communicator, n_fe_indices);
1789 *  
1790 *   pcout << " Frequencies of poly. degrees:";
1791 *   for (unsigned int i = 0; i < fe_collection.size(); ++i)
1792 *   if (n_fe_indices[i] > 0)
1793 *   pcout << ' ' << fe_collection[i].degree << ':' << n_fe_indices[i];
1794 *   pcout << std::endl;
1795 *   }
1796 *   }
1797 *  
1798 *  
1799 *  
1800 * @endcode
1801 *
1802 *
1803 * <a name="step_75-LaplaceProblemsolve_system"></a>
1804 * <h4>LaplaceProblem::solve_system</h4>
1805 *
1806
1807 *
1808 * The scaffold around the solution is similar to the one of @ref step_40 "step-40". We
1809 * prepare a vector that matches the requirements of MatrixFree and collect
1810 * the locally-relevant degrees of freedoms we solved the equation system. The
1811 * solution happens with the function introduced earlier.
1812 *
1813 * @code
1814 *   template <int dim>
1815 *   void LaplaceProblem<dim>::solve_system()
1816 *   {
1817 *   TimerOutput::Scope t(computing_timer, "solve system");
1818 *  
1819 *   LinearAlgebra::distributed::Vector<double> completely_distributed_solution;
1820 *   laplace_operator.initialize_dof_vector(completely_distributed_solution);
1821 *  
1822 *   SolverControl solver_control(system_rhs.size(),
1823 *   prm.tolerance_factor * system_rhs.l2_norm());
1824 *  
1825 *   solve_with_gmg(solver_control,
1826 *   laplace_operator,
1827 *   completely_distributed_solution,
1828 *   system_rhs,
1829 *   prm.mg_data,
1830 *   mapping_collection,
1831 *   dof_handler,
1832 *   quadrature_collection);
1833 *  
1834 *   pcout << " Solved in " << solver_control.last_step() << " iterations."
1835 *   << std::endl;
1836 *  
1837 *   constraints.distribute(completely_distributed_solution);
1838 *  
1839 *   locally_relevant_solution.copy_locally_owned_data_from(
1840 *   completely_distributed_solution);
1841 *   locally_relevant_solution.update_ghost_values();
1842 *   }
1843 *  
1844 *  
1845 *  
1846 * @endcode
1847 *
1848 *
1849 * <a name="step_75-LaplaceProblemcompute_indicators"></a>
1850 * <h4>LaplaceProblem::compute_indicators</h4>
1851 *
1852
1853 *
1854 * This function contains only a part of the typical <code>refine_grid</code>
1855 * function from other tutorials and is new in that sense. Here, we will only
1856 * calculate all indicators for adaptation with actually refining the grid. We
1857 * do this for the purpose of writing all indicators to the file system, so we
1858 * store them for later.
1859 *
1860
1861 *
1862 * Since we are dealing the an elliptic problem, we will make use of the
1863 * KellyErrorEstimator again, but with a slight difference. Modifying the
1864 * scaling factor of the underlying face integrals to be dependent on the
1865 * actual polynomial degree of the neighboring elements is favorable in
1866 * hp-adaptive applications @cite davydov2017hp. We can do this by specifying
1867 * the very last parameter from the additional ones you notices. The others
1868 * are actually just the defaults.
1869 *
1870
1871 *
1872 * For the purpose of hp-adaptation, we will calculate smoothness estimates
1873 * with the strategy presented in the tutorial introduction and use the
1874 * implementation in SmoothnessEstimator::Legendre. In the Parameters struct,
1875 * we set the minimal polynomial degree to 2 as it seems that the smoothness
1876 * estimation algorithms have trouble with linear elements.
1877 *
1878 * @code
1879 *   template <int dim>
1880 *   void LaplaceProblem<dim>::compute_indicators()
1881 *   {
1882 *   TimerOutput::Scope t(computing_timer, "compute indicators");
1883 *  
1884 *   estimated_error_per_cell.grow_or_shrink(triangulation.n_active_cells());
1886 *   dof_handler,
1887 *   face_quadrature_collection,
1888 *   std::map<types::boundary_id, const Function<dim> *>(),
1889 *   locally_relevant_solution,
1890 *   estimated_error_per_cell,
1891 *   /*component_mask=*/ComponentMask(),
1892 *   /*coefficients=*/nullptr,
1893 *   /*n_threads=*/numbers::invalid_unsigned_int,
1894 *   /*subdomain_id=*/numbers::invalid_subdomain_id,
1895 *   /*material_id=*/numbers::invalid_material_id,
1896 *   /*strategy=*/
1898 *  
1899 *   hp_decision_indicators.grow_or_shrink(triangulation.n_active_cells());
1901 *   dof_handler,
1902 *   locally_relevant_solution,
1903 *   hp_decision_indicators);
1904 *   }
1905 *  
1906 *  
1907 *  
1908 * @endcode
1909 *
1910 *
1911 * <a name="step_75-LaplaceProblemadapt_resolution"></a>
1912 * <h4>LaplaceProblem::adapt_resolution</h4>
1913 *
1914
1915 *
1916 * With the previously calculated indicators, we will finally flag all cells
1917 * for adaptation and also execute refinement in this function. As in previous
1918 * tutorials, we will use the "fixed number" strategy, but now for
1919 * hp-adaptation.
1920 *
1921 * @code
1922 *   template <int dim>
1923 *   void LaplaceProblem<dim>::adapt_resolution()
1924 *   {
1925 *   TimerOutput::Scope t(computing_timer, "adapt resolution");
1926 *  
1927 * @endcode
1928 *
1929 * First, we will set refine and coarsen flags based on the error estimates
1930 * on each cell. There is nothing new here.
1931 *
1932
1933 *
1934 * We will use general refine and coarsen fractions that have been
1935 * elaborated in the other deal.II tutorials: using the fixed number
1936 * strategy, we will flag 30% of all cells for refinement and 3% for
1937 * coarsening, as provided in the Parameters struct.
1938 *
1939 * @code
1941 *   triangulation,
1942 *   estimated_error_per_cell,
1943 *   prm.refine_fraction,
1944 *   prm.coarsen_fraction);
1945 *  
1946 * @endcode
1947 *
1948 * Next, we will make all adjustments for hp-adaptation. We want to refine
1949 * and coarsen those cells flagged in the previous step, but need to decide
1950 * if we would like to do it by adjusting the grid resolution or the
1951 * polynomial degree.
1952 *
1953
1954 *
1955 * The next function call sets future FE indices according to the previously
1956 * calculated smoothness indicators as p-adaptation indicators. These
1957 * indices will only be set on those cells that have refine or coarsen flags
1958 * assigned.
1959 *
1960
1961 *
1962 * For the p-adaptation fractions, we will take an educated guess. Since we
1963 * only expect a single singularity in our scenario, i.e., in the origin of
1964 * the domain, and a smooth solution anywhere else, we would like to
1965 * strongly prefer to use p-adaptation over h-adaptation. This reflects in
1966 * our choice of a fraction of 90% for both p-refinement and p-coarsening.
1967 *
1968 * @code
1970 *   hp_decision_indicators,
1971 *   prm.p_refine_fraction,
1972 *   prm.p_coarsen_fraction);
1973 *  
1974 * @endcode
1975 *
1976 * After setting all indicators, we will remove those that exceed the
1977 * specified limits of the provided level ranges in the Parameters struct.
1978 * This limitation naturally arises for p-adaptation as the number of
1979 * supplied finite elements is limited. In addition, we registered a custom
1980 * hierarchy for p-adaptation in the constructor. Now, we need to do this
1981 * manually in the h-adaptive context like in @ref step_31 "step-31".
1982 *
1983
1984 *
1985 * We will iterate over all cells on the designated min and max levels and
1986 * remove the corresponding flags. As an alternative, we could also flag
1987 * these cells for p-adaptation by setting future FE indices accordingly
1988 * instead of simply clearing the refine and coarsen flags.
1989 *
1990 * @code
1991 *   Assert(triangulation.n_levels() >= prm.min_h_level + 1 &&
1992 *   triangulation.n_levels() <= prm.max_h_level + 1,
1993 *   ExcInternalError());
1994 *  
1995 *   if (triangulation.n_levels() > prm.max_h_level)
1996 *   for (const auto &cell :
1997 *   triangulation.active_cell_iterators_on_level(prm.max_h_level))
1998 *   cell->clear_refine_flag();
1999 *  
2000 *   for (const auto &cell :
2001 *   triangulation.active_cell_iterators_on_level(prm.min_h_level))
2002 *   cell->clear_coarsen_flag();
2003 *  
2004 * @endcode
2005 *
2006 * At this stage, we have both the future FE indices and the classic refine
2007 * and coarsen flags set. The latter will be interpreted by
2008 * Triangulation::execute_coarsening_and_refinement() for h-adaptation, and
2009 * our previous modification ensures that the resulting Triangulation stays
2010 * within the specified level range.
2011 *
2012
2013 *
2014 * Now, we would like to only impose one type of adaptation on cells, which
2015 * is what the next function will sort out for us. In short, on cells which
2016 * have both types of indicators assigned, we will favor the p-adaptation
2017 * one and remove the h-adaptation one.
2018 *
2019 * @code
2020 *   hp::Refinement::choose_p_over_h(dof_handler);
2021 *  
2022 * @endcode
2023 *
2024 * In the end, we are left to execute coarsening and refinement. Here, not
2025 * only the grid will be updated, but also all previous future FE indices
2026 * will become active.
2027 *
2028
2029 *
2030 * Remember that we have attached functions to triangulation signals in the
2031 * constructor, will be triggered in this function call. So there is even
2032 * more happening: weighted repartitioning will be performed to ensure load
2033 * balancing, as well as we will limit the difference of p-levels between
2034 * neighboring cells.
2035 *
2036 * @code
2037 *   triangulation.execute_coarsening_and_refinement();
2038 *   }
2039 *  
2040 *  
2041 *  
2042 * @endcode
2043 *
2044 *
2045 * <a name="step_75-LaplaceProblemoutput_results"></a>
2046 * <h4>LaplaceProblem::output_results</h4>
2047 *
2048
2049 *
2050 * Writing results to the file system in parallel applications works exactly
2051 * like in @ref step_40 "step-40". In addition to the data containers that we prepared
2052 * throughout the tutorial, we would also like to write out the polynomial
2053 * degree of each finite element on the grid as well as the subdomain each
2054 * cell belongs to. We prepare necessary containers for this in the scope of
2055 * this function.
2056 *
2057 * @code
2058 *   template <int dim>
2059 *   void LaplaceProblem<dim>::output_results(const unsigned int cycle)
2060 *   {
2061 *   TimerOutput::Scope t(computing_timer, "output results");
2062 *  
2063 *   Vector<float> fe_degrees(triangulation.n_active_cells());
2064 *   for (const auto &cell : dof_handler.active_cell_iterators())
2065 *   if (cell->is_locally_owned())
2066 *   fe_degrees(cell->active_cell_index()) = cell->get_fe().degree;
2067 *  
2068 *   Vector<float> subdomain(triangulation.n_active_cells());
2069 *   for (auto &subd : subdomain)
2071 *  
2072 *   DataOut<dim> data_out;
2073 *   data_out.attach_dof_handler(dof_handler);
2074 *   data_out.add_data_vector(locally_relevant_solution, "solution");
2075 *   data_out.add_data_vector(fe_degrees, "fe_degree");
2076 *   data_out.add_data_vector(subdomain, "subdomain");
2077 *   data_out.add_data_vector(estimated_error_per_cell, "error");
2078 *   data_out.add_data_vector(hp_decision_indicators, "hp_indicator");
2079 *   data_out.build_patches(mapping_collection);
2080 *  
2081 *   data_out.write_vtu_with_pvtu_record(
2082 *   "./", "solution", cycle, mpi_communicator, 2, 1);
2083 *   }
2084 *  
2085 *  
2086 *  
2087 * @endcode
2088 *
2089 *
2090 * <a name="step_75-LaplaceProblemrun"></a>
2091 * <h4>LaplaceProblem::run</h4>
2092 *
2093
2094 *
2095 * The actual run function again looks very familiar to @ref step_40 "step-40". The only
2096 * addition is the bracketed section that precedes the actual cycle loop.
2097 * Here, we will pre-calculate the Legendre transformation matrices. In
2098 * general, these will be calculated on the fly via lazy allocation whenever a
2099 * certain matrix is needed. For timing purposes however, we would like to
2100 * calculate them all at once before the actual time measurement begins. We
2101 * will thus designate their calculation to their own scope.
2102 *
2103 * @code
2104 *   template <int dim>
2105 *   void LaplaceProblem<dim>::run()
2106 *   {
2107 *   pcout << "Running with Trilinos on "
2108 *   << Utilities::MPI::n_mpi_processes(mpi_communicator)
2109 *   << " MPI rank(s)..." << std::endl;
2110 *  
2111 *   {
2112 *   pcout << "Calculating transformation matrices..." << std::endl;
2113 *   TimerOutput::Scope t(computing_timer, "calculate transformation");
2114 *   legendre->precalculate_all_transformation_matrices();
2115 *   }
2116 *  
2117 *   for (unsigned int cycle = 0; cycle < prm.n_cycles; ++cycle)
2118 *   {
2119 *   pcout << "Cycle " << cycle << ':' << std::endl;
2120 *  
2121 *   if (cycle == 0)
2122 *   initialize_grid();
2123 *   else
2124 *   adapt_resolution();
2125 *  
2126 *   setup_system();
2127 *  
2128 *   print_diagnostics();
2129 *  
2130 *   solve_system();
2131 *  
2132 *   compute_indicators();
2133 *  
2134 *   if (Utilities::MPI::n_mpi_processes(mpi_communicator) <= 32)
2135 *   output_results(cycle);
2136 *  
2137 *   computing_timer.print_summary();
2138 *   computing_timer.reset();
2139 *  
2140 *   pcout << std::endl;
2141 *   }
2142 *   }
2143 *   } // namespace Step75
2144 *  
2145 *  
2146 *  
2147 * @endcode
2148 *
2149 *
2150 * <a name="step_75-main"></a>
2151 * <h4>main()</h4>
2152 *
2153
2154 *
2155 * The final function is the <code>main</code> function that will ultimately
2156 * create and run a LaplaceOperator instantiation. Its structure is similar to
2157 * most other tutorial programs.
2158 *
2159 * @code
2160 *   int main(int argc, char *argv[])
2161 *   {
2162 *   try
2163 *   {
2164 *   using namespace dealii;
2165 *   using namespace Step75;
2166 *  
2167 *   Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
2168 *  
2169 *   Parameters prm;
2170 *   LaplaceProblem<2> laplace_problem(prm);
2171 *   laplace_problem.run();
2172 *   }
2173 *   catch (std::exception &exc)
2174 *   {
2175 *   std::cerr << std::endl
2176 *   << std::endl
2177 *   << "----------------------------------------------------"
2178 *   << std::endl;
2179 *   std::cerr << "Exception on processing: " << std::endl
2180 *   << exc.what() << std::endl
2181 *   << "Aborting!" << std::endl
2182 *   << "----------------------------------------------------"
2183 *   << std::endl;
2184 *  
2185 *   return 1;
2186 *   }
2187 *   catch (...)
2188 *   {
2189 *   std::cerr << std::endl
2190 *   << std::endl
2191 *   << "----------------------------------------------------"
2192 *   << std::endl;
2193 *   std::cerr << "Unknown exception!" << std::endl
2194 *   << "Aborting!" << std::endl
2195 *   << "----------------------------------------------------"
2196 *   << std::endl;
2197 *   return 1;
2198 *   }
2199 *  
2200 *   return 0;
2201 *   }
2202 * @endcode
2203<a name="step_75-Results"></a><h1>Results</h1>
2204
2205
2206When you run the program with the given parameters on four processes in
2207release mode, your terminal output should look like this:
2208@code
2209Running with Trilinos on 4 MPI rank(s)...
2210Calculating transformation matrices...
2211Cycle 0:
2212 Number of active cells: 3072
2213 by partition: 768 768 768 768
2214 Number of degrees of freedom: 12545
2215 by partition: 3201 3104 3136 3104
2216 Number of constraints: 542
2217 by partition: 165 74 138 165
2218 Frequencies of poly. degrees: 2:3072
2219 Solved in 7 iterations.
2220
2221
2222+---------------------------------------------+------------+------------+
2223| Total wallclock time elapsed since start | 0.172s | |
2224| | | |
2225| Section | no. calls | wall time | % of total |
2226+---------------------------------+-----------+------------+------------+
2227| calculate transformation | 1 | 0.0194s | 11% |
2228| compute indicators | 1 | 0.00676s | 3.9% |
2229| initialize grid | 1 | 0.011s | 6.4% |
2230| output results | 1 | 0.0343s | 20% |
2231| setup system | 1 | 0.00839s | 4.9% |
2232| solve system | 1 | 0.0896s | 52% |
2233+---------------------------------+-----------+------------+------------+
2234
2235
2236Cycle 1:
2237 Number of active cells: 3351
2238 by partition: 875 761 843 872
2239 Number of degrees of freedom: 18228
2240 by partition: 4535 4735 4543 4415
2241 Number of constraints: 1202
2242 by partition: 303 290 326 283
2243 Frequencies of poly. degrees: 2:2522 3:829
2244 Solved in 7 iterations.
2245
2246
2247+---------------------------------------------+------------+------------+
2248| Total wallclock time elapsed since start | 0.165s | |
2249| | | |
2250| Section | no. calls | wall time | % of total |
2251+---------------------------------+-----------+------------+------------+
2252| adapt resolution | 1 | 0.00473s | 2.9% |
2253| compute indicators | 1 | 0.00764s | 4.6% |
2254| output results | 1 | 0.0243s | 15% |
2255| setup system | 1 | 0.00662s | 4% |
2256| solve system | 1 | 0.121s | 74% |
2257+---------------------------------+-----------+------------+------------+
2258
2259
2260...
2261
2262
2263Cycle 7:
2264 Number of active cells: 5610
2265 by partition: 1324 1483 1482 1321
2266 Number of degrees of freedom: 82047
2267 by partition: 21098 19960 20111 20878
2268 Number of constraints: 14357
2269 by partition: 3807 3229 3554 3767
2270 Frequencies of poly. degrees: 2:1126 3:1289 4:2725 5:465 6:5
2271 Solved in 7 iterations.
2272
2273
2274+---------------------------------------------+------------+------------+
2275| Total wallclock time elapsed since start | 1.83s | |
2276| | | |
2277| Section | no. calls | wall time | % of total |
2278+---------------------------------+-----------+------------+------------+
2279| adapt resolution | 1 | 0.00834s | 0.46% |
2280| compute indicators | 1 | 0.0178s | 0.97% |
2281| output results | 1 | 0.0434s | 2.4% |
2282| setup system | 1 | 0.0165s | 0.9% |
2283| solve system | 1 | 1.74s | 95% |
2284+---------------------------------+-----------+------------+------------+
2285@endcode
2286
2287When running the code with more processes, you will notice slight
2288differences in the number of active cells and degrees of freedom. This
2289is due to the fact that solver and preconditioner depend on the
2290partitioning of the problem, which might yield to slight differences of
2291the solution in the last digits and ultimately yields to different
2292adaptation behavior.
2293
2294Furthermore, the number of iterations for the solver stays about the
2295same in all cycles despite hp-adaptation, indicating the robustness of
2296the proposed algorithms and promising good scalability for even larger
2297problem sizes and on more processes.
2298
2299Let us have a look at the graphical output of the program. After all
2300refinement cycles in the given parameter configuration, the actual
2301discretized function space looks like the following with its
2302partitioning on twelve processes on the left and the polynomial degrees
2303of finite elements on the right. In the left picture, each color
2304represents a unique subdomain. In the right picture, the lightest color
2305corresponds to the polynomial degree two and the darkest one corresponds
2306to degree six:
2307
2308<div class="twocolumn" style="width: 80%; text-align: center;">
2309 <div>
2310 <img src="https://www.dealii.org/images/steps/developer/step-75.subdomains-07.svg"
2311 alt="Partitioning after seven refinements.">
2312 </div>
2313 <div>
2314 <img src="https://www.dealii.org/images/steps/developer/step-75.fedegrees-07.svg"
2315 alt="Local approximation degrees after seven refinements.">
2316 </div>
2317</div>
2318
2319
2320
2321<a name="step-75-extensions"></a>
2322<a name="step_75-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
2323
2324
2325This tutorial shows only one particular way how to use parallel
2326hp-adaptive finite element methods. In the following paragraphs, you
2327will get to know which alternatives are possible. Most of these
2328extensions are already part of https://github.com/marcfehling/hpbox/,
2329which provides you with implementation examples that you can play
2330around with.
2331
2332
2333<a name="step_75-Differenthpdecisionstrategies"></a><h4>Different hp-decision strategies</h4>
2334
2335
2336The deal.II library offers multiple strategies to decide which type of
2337adaptation to impose on cells: either adjust the grid resolution or
2338change the polynomial degree. We only presented the <i>Legendre
2339coefficient decay</i> strategy in this tutorial, while @ref step_27 "step-27"
2340demonstrated the <i>Fourier</i> equivalent of the same idea.
2341
2342See the "possibilities for extensions" section of @ref step_27 "step-27" for an
2343overview over these strategies, or the corresponding documentation
2344for a detailed description.
2345
2346There, another strategy is mentioned that has not been shown in any
2347tutorial so far: the strategy based on <i>refinement history</i>. The
2348usage of this method for parallel distributed applications is more
2349tricky than the others, so we will highlight the challenges that come
2350along with it. We need information about the final state of refinement
2351flags, and we need to transfer the solution across refined meshes. For
2352the former, we need to attach the hp::Refinement::predict_error()
2353function to the Triangulation::Signals::post_p4est_refinement signal in
2354a way that it will be called <i>after</i> the
2355hp::Refinement::limit_p_level_difference() function. At this stage, all
2356refinement flags and future FE indices are terminally set and a reliable
2357prediction of the error is possible. The predicted error then needs to
2358be transferred across refined meshes with the aid of
2359parallel::distributed::CellDataTransfer.
2360
2361Try implementing one of these strategies into this tutorial and observe
2362the subtle changes to the results. You will notice that all strategies
2363are capable of identifying the singularities near the reentrant corners
2364and will perform @f$h@f$-refinement in these regions, while preferring
2365@f$p@f$-refinement in the bulk domain. A detailed comparison of these
2366strategies is presented in @cite fehling2020 .
2367
2368
2369<a name="step_75-Solvewithmatrixbasedmethods"></a><h4>Solve with matrix-based methods</h4>
2370
2371
2372This tutorial focuses solely on matrix-free strategies. All hp-adaptive
2373algorithms however also work with matrix-based approaches in the
2374parallel distributed context.
2375
2376To create a system matrix, you can either use the
2377LaplaceOperator::get_system_matrix() function, or use an
2378<code>assemble_system()</code> function similar to the one of @ref step_27 "step-27".
2379You can then pass the system matrix to the solver as usual.
2380
2381You can time the results of both matrix-based and matrix-free
2382implementations, quantify the speed-up, and convince yourself which
2383variant is faster.
2384
2385
2386<a name="step_75-Multigridvariants"></a><h4>Multigrid variants</h4>
2387
2388
2389For sake of simplicity, we have restricted ourselves to a single type of
2390coarse-grid solver (CG with AMG), smoother (Chebyshev smoother with
2391point Jacobi preconditioner), and geometric-coarsening scheme (global
2392coarsening) within the multigrid algorithm. Feel free to try out
2393alternatives and investigate their performance and robustness.
2394 *
2395 *
2396<a name="step_75-PlainProg"></a>
2397<h1> The plain program</h1>
2398@include "step-75.cc"
2399*/
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
void reinit(const Triangulation< dim, spacedim > &tria)
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
Definition fe_q.h:554
static void estimate(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim - 1 > &quadrature, const std::map< types::boundary_id, const Function< spacedim, Number > * > &neumann_bc, const ReadVector< Number > &solution, Vector< float > &error, const ComponentMask &component_mask={}, const Function< spacedim > *coefficients=nullptr, const unsigned int n_threads=numbers::invalid_unsigned_int, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id, const types::material_id material_id=numbers::invalid_material_id, const Strategy strategy=cell_diameter_over_24)
void initialize(const MGLevelObject< MatrixType2 > &matrices, const typename PreconditionerType::AdditionalData &additional_data=typename PreconditionerType::AdditionalData())
void initialize_dof_vector(VectorType &vec, const unsigned int dof_handler_index=0) const
void reinit(const MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
Definition point.h:111
void solve(const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)
@ wall_times
Definition timer.h:651
virtual types::subdomain_id locally_owned_subdomain() const
unsigned int n_active_cells() const
void coarsen_global(const unsigned int times=1)
void refine_global(const unsigned int times=1)
unsigned int n_levels() const
virtual void execute_coarsening_and_refinement()
Signals signals
Definition tria.h:2513
unsigned int size() const
Definition collection.h:308
static WeightingFunction ndofs_weighting(const std::pair< float, float > &coefficients)
virtual types::global_cell_index n_global_active_cells() const override
Definition tria_base.cc:151
unsigned int n_locally_owned_active_cells() const
Definition tria_base.cc:131
Point< 2 > second
Definition grid_out.cc:4624
Point< 2 > first
Definition grid_out.cc:4623
unsigned int level
Definition grid_out.cc:4626
IteratorRange< active_cell_iterator > active_cell_iterators() const
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
#define AssertThrow(cond, exc)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:564
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints={}, const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
@ update_gradients
Shape function gradients.
#define DEAL_II_NOT_IMPLEMENTED()
IndexSet extract_locally_relevant_dofs(const DoFHandler< dim, spacedim > &dof_handler)
std::array< double, dim > to_spherical(const Point< dim > &point)
void hyper_L(Triangulation< dim > &tria, const double left=-1., const double right=1., const bool colorize=false)
void subdivided_hyper_L(Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &bottom_left, const Point< dim > &top_right, const std::vector< int > &n_cells_to_remove)
void refine(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold, const unsigned int max_to_mark=numbers::invalid_unsigned_int)
void coarsen(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold)
void scale(const double scaling_factor, Triangulation< dim, spacedim > &triangulation)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
std::vector< std::shared_ptr< const Triangulation< dim, spacedim > > > create_geometric_coarsening_sequence(const Triangulation< dim, spacedim > &tria)
unsigned int create_next_polynomial_coarsening_degree(const unsigned int degree, const PolynomialCoarseningSequenceType &p_sequence)
std::vector< unsigned int > create_polynomial_coarsening_sequence(const unsigned int max_degree, const PolynomialCoarseningSequenceType &p_sequence)
void compute_matrix(const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free, const AffineConstraints< Number > &constraints, MatrixType &matrix, const std::function< void(FEEvaluation< dim, fe_degree, n_q_points_1d, n_components, Number, VectorizedArrayType > &)> &cell_operation, const unsigned int dof_no=0, const unsigned int quad_no=0, const unsigned int first_selected_component=0)
void compute_diagonal(const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free, VectorType &diagonal_global, const std::function< void(FEEvaluation< dim, fe_degree, n_q_points_1d, n_components, Number, VectorizedArrayType > &)> &cell_operation, const unsigned int dof_no=0, const unsigned int quad_no=0, const unsigned int first_selected_component=0)
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 > b(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
FESeries::Legendre< dim, spacedim > default_fe_series(const hp::FECollection< dim, spacedim > &fe_collection, const unsigned int component=numbers::invalid_unsigned_int)
void coefficient_decay(FESeries::Legendre< dim, spacedim > &fe_legendre, const DoFHandler< dim, spacedim > &dof_handler, const VectorType &solution, Vector< float > &smoothness_indicators, const VectorTools::NormType regression_strategy=VectorTools::Linfty_norm, const double smallest_abs_coefficient=1e-10, const bool only_flagged_cells=false)
void partition(const SparsityPattern &sparsity_pattern, const unsigned int n_partitions, std::vector< unsigned int > &partition_indices, const Partitioner partitioner=Partitioner::metis)
void free(T *&pointer)
Definition cuda.h:96
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int n_mpi_processes(const MPI_Comm mpi_communicator)
Definition mpi.cc:92
T max(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
Definition mpi.cc:107
std::vector< T > gather(const MPI_Comm comm, const T &object_to_send, const unsigned int root_process=0)
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask={})
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
bool limit_p_level_difference(const DoFHandler< dim, spacedim > &dof_handler, const unsigned int max_difference=1, const unsigned int contains_fe_index=0)
void predict_error(const DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &error_indicators, Vector< Number > &predicted_errors, const double gamma_p=std::sqrt(0.4), const double gamma_h=2., const double gamma_n=1.)
void p_adaptivity_fixed_number(const DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const double p_refine_fraction=0.5, const double p_coarsen_fraction=0.5, const ComparisonFunction< std_cxx20::type_identity_t< Number > > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< std_cxx20::type_identity_t< Number > > &compare_coarsen=std::less_equal< Number >())
Definition hp.h:117
int(& functions)(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
Definition mg.h:81
const types::material_id invalid_material_id
Definition types.h:277
const types::subdomain_id invalid_subdomain_id
Definition types.h:341
static const unsigned int invalid_unsigned_int
Definition types.h:220
void refine_and_coarsen_fixed_number(::Triangulation< dim, spacedim > &tria, const ::Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const types::global_cell_index max_n_cells=std::numeric_limits< types::global_cell_index >::max())
double legendre(unsigned int l, double x)
Definition cmath.h:65
STL namespace.
::VectorizedArray< Number, width > min(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
Definition types.h:32
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
boost::signals2::signal< void()> post_p4est_refinement
Definition tria.h:2448