 |
DUBeat
1.0.1
High-order discontinuous Galerkin methods and applications to cardiac electrophysiology
|
|
DUBeat
1.0.1
High-order discontinuous Galerkin methods and applications to cardiac electrophysiology
|
Class to solve the heat equation using the Discontinuous Galerkin method. More...
#include <heat_dg.hpp>
Public Member Functions | |
| HeatDG () | |
| Constructor. More... | |
 Public Member Functions inherited from ModelDG_t< basis > | |
| ModelDG_t (std::string model_name) | |
| Constructor. More... | |
| ModelDG_t (ModelDG_t< basis > &ModelDG_t)=default | |
| Default copy constructor. More... | |
| ModelDG_t (const ModelDG_t< basis > &ModelDG_t)=default | |
| Default const copy constructor. More... | |
| ModelDG_t (ModelDG_t< basis > &&ModelDG_t)=default | |
| Default move constructor. More... | |
| virtual | ~ModelDG_t ()=default |
| Destructor. More... | |
| Public Member Functions inherited from ModelDG< basis > | |
| ModelDG (std::string model_name) | |
| Constructor. More... | |
| ModelDG (ModelDG< basis > &ModelDG)=default | |
| Default copy constructor. More... | |
| ModelDG (const ModelDG< basis > &ModelDG)=default | |
| Default const copy constructor. More... | |
| ModelDG (ModelDG< basis > &&ModelDG)=default | |
| Default move constructor. More... | |
| unsigned int | get_dofs_per_cell () const |
| Return the number of degrees of freedom per element. More... | |
| virtual | ~ModelDG ()=default |
| Destructor. More... | |
Private Member Functions | |
| void | assemble_system () override |
| Assembly of the linear system. More... | |
Additional Inherited Members | |
| Protected Member Functions inherited from ModelDG_t< basis > | |
| virtual void | declare_parameters (lifex::ParamHandler ¶ms) const override |
| Override for declaration of additional parameters. More... | |
| virtual void | parse_parameters (lifex::ParamHandler ¶ms) override |
| Override to parse additional parameters. More... | |
| virtual void | time_initialization () |
| Setup for the time-dependent problems at the initial time-step. More... | |
| virtual void | update_time () |
| To perform the time increment. More... | |
| virtual void | intermediate_error_print (const lifex::LinAlg::MPI::Vector &solution_owned, const lifex::LinAlg::MPI::Vector &solution_ex_owned, const std::shared_ptr< dealii::Function< lifex::dim >> &u_ex, const char *solution_name=(char *)"u") |
| Computation of the \(L^\infty\) error at an intermediate time-step. More... | |
| void | run () override |
| Override for the simulation run. More... | |
| Protected Member Functions inherited from ModelDG< basis > | |
| virtual void | setup_system () |
| Setup of the problem before the resolution. More... | |
| void | make_sparsity_pattern (const DoFHandlerDG< basis > &dof, dealii::DynamicSparsityPattern &sparsity, const dealii::AffineConstraints< double > &constraints=dealii::AffineConstraints< double >(), const bool keep_constrained_dofs=true, const dealii::types::subdomain_id subdomain_id=dealii::numbers::invalid_subdomain_id) |
| Creation of the sparsity pattern to assign to the system matrix before assembling. More... | |
| void | initialize_solution (lifex::LinAlg::MPI::Vector &solution_owned, lifex::LinAlg::MPI::Vector &solution) |
| To inizialize the solutions using the deal.II reinit. More... | |
| void | create_mesh () |
| Load the mesh from the default path. More... | |
| void | create_mesh (std::string mesh_path) |
| Load the mesh from a user-defined path. More... | |
| void | solve_system () |
| System solving. More... | |
| void | compute_errors (const lifex::LinAlg::MPI::Vector &solution_owned, const lifex::LinAlg::MPI::Vector &solution_ex_owned, const std::shared_ptr< dealii::Function< lifex::dim >> &u_ex, const std::shared_ptr< dealii::Function< lifex::dim >> &grad_u_ex, const char *solution_name=(char *)"u") const |
| To compute the \(L^\infty\) error, the \(L^2\) error, the \(H^1\) error and the \(DG\) error at the end of system solving, it exploits the DGComputeErrors<basis> class. More... | |
| virtual void | output_results (std::string output_name="solution") const |
| Output of results. More... | |
| virtual void | conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned) |
| To convert a discretized solution from modal to nodal basis (does nothing if problem is already in nodal basis), in-place version. More... | |
| virtual lifex::LinAlg::MPI::Vector | conversion_to_fem (const lifex::LinAlg::MPI::Vector &sol_owned) const |
| To convert a discretized solution from modal to nodal basis (does nothing if problem is already in nodal basis), const version. More... | |
| virtual void | conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem=1, const double scaling_factor=1) const |
| virtual void | conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned) |
| To convert a discretized solution in Dubiner basis (only for problems using Dubiner basis). More... | |
| virtual lifex::LinAlg::MPI::Vector | conversion_to_dub (const lifex::LinAlg::MPI::Vector &sol_owned) const |
| To convert a discretized solution in Dubiner basis (only for problems using Dubiner basis). More... | |
| virtual void | conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem=1, const double scaling_factor=1) const |
| void | discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned) |
| Conversion of an analytical solution from FEM to basis coefficients. More... | |
| void | conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned) |
| Conversion of a discretized solution from Dubiner coefficients to FEM coefficients. More... | |
| lifex::LinAlg::MPI::Vector | conversion_to_fem (const lifex::LinAlg::MPI::Vector &sol_owned) const |
| Conversion to FEM coefficients, const version. More... | |
| void | conversion_to_fem (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem, const double scaling_factor) const |
| void | conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned) |
| Conversion of a discretized solution from FEM coefficients to Dubiner coefficients. More... | |
| lifex::LinAlg::MPI::Vector | conversion_to_dub (const lifex::LinAlg::MPI::Vector &sol_owned) const |
| Conversion to DUB coefficients, const version. More... | |
| void | conversion_to_dub (lifex::LinAlg::MPI::Vector &sol_owned, const std::string fem_file_path, const unsigned int degree_fem, const double scaling_factor) const |
| void | discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned) |
| Conversion of an analytical solution from FEM to basis coefficients. More... | |
| void | discretize_analytical_solution (const std::shared_ptr< dealii::Function< lifex::dim >> &u_analytical, lifex::LinAlg::MPI::Vector &sol_owned) |
| Conversion of an analytical solution from FEM to basis coefficients. More... | |
| Protected Attributes inherited from ModelDG_t< basis > | |
| double | prm_time_init |
| Initial time. More... | |
| double | prm_time_final |
| Final time. More... | |
| double | prm_time_step |
| Time-step amplitude. More... | |
| unsigned int | prm_bdf_order |
| BDF order. More... | |
| double | time |
| Current time. More... | |
| unsigned int | timestep_number |
| Current time-step number. More... | |
| lifex::utils::BDFHandler< lifex::LinAlg::MPI::Vector > | bdf_handler |
| BDF time advancing handler. More... | |
| lifex::LinAlg::MPI::Vector | solution_bdf |
| BDF solution, with ghost entries. More... | |
| lifex::LinAlg::MPI::Vector | solution_ext |
| BDF extrapolated solution, with ghost entries. More... | |
| Protected Attributes inherited from ModelDG< basis > | |
| const std::string | model_name |
| Name of the class/problem. More... | |
| unsigned int | prm_fe_degree |
| Polynomials degree. More... | |
| unsigned int | prm_n_refinements |
| Mesh refinement level (>=1). More... | |
| double | prm_penalty_coeff |
| DG Penalty coefficient. More... | |
| double | prm_stability_coeff |
| DG stabilty coefficient. More... | |
| double | scaling_factor = 1 |
| Scaling factor. More... | |
| std::shared_ptr< lifex::utils::MeshHandler > | triangulation |
| Triangulation (internal use for useful already implemented methods). More... | |
| unsigned int | dofs_per_cell |
| Number of degrees of freedom per cell. More... | |
| DoFHandlerDG< basis > | dof_handler |
| DoFHandler (internal use for useful already implemented methods). More... | |
| std::shared_ptr< DUBFEMHandler< basis > > | dub_fem_values |
| Member used for conversions between analytical, nodal and modal representations of the solutions. More... | |
| std::unique_ptr< AssembleDG< basis > > | assemble |
| Matrix assembler. More... | |
| lifex::utils::LinearSolverHandler< lifex::LinAlg::MPI::Vector > | linear_solver |
| Linear solver handler. More... | |
| lifex::utils::PreconditionerHandler | preconditioner |
| Preconditioner handler. More... | |
| lifex::LinAlg::MPI::SparseMatrix | matrix |
| Distributed matrix of the linear system. More... | |
| lifex::LinAlg::MPI::Vector | rhs |
| Distributed right hand side vector of the linear system. More... | |
| lifex::LinAlg::MPI::Vector | solution_owned |
| Distributed solution vector, without ghost entries. More... | |
| lifex::LinAlg::MPI::Vector | solution |
| Distributed solution vector, with ghost entries. More... | |
| lifex::LinAlg::MPI::Vector | solution_ex_owned |
| Distributed exact solution vector, without ghost entries. More... | |
| lifex::LinAlg::MPI::Vector | solution_ex |
| Distributed exact solution vector, without ghost entries. More... | |
| std::shared_ptr< lifex::utils::FunctionDirichlet > | u_ex |
| Pointer to exact solution function. More... | |
| std::shared_ptr< dealii::Function< lifex::dim > > | grad_u_ex |
| Pointer to exact gradient solution Function. More... | |
| std::shared_ptr< dealii::Function< lifex::dim > > | f_ex |
| Known forcing term. More... | |
| std::shared_ptr< dealii::Function< lifex::dim > > | g_n |
| Neumann boundary conditions. More... | |
Class to solve the heat equation using the Discontinuous Galerkin method.
\[ \begin{aligned} \frac{\partial u}{\partial t} -\Delta u &= f, & \quad & \text{in } \Omega = (-1, 1)^d \times (0, T], \\ \frac{\partial u}{\partial n} &= g, & \quad & \text{on } \partial\Omega \times (0, T], \\ u &= u_\mathrm{ex}, & \quad & \text{in } \Omega \times \{ t = 0 \}. \end{aligned} \]
In particular, it can be solved using the basis functions the FEM basis (basis=dealii::FE_SimplexDGP<lifex::dim>) or the Dubiner basis (basis=DUBValues<lifex::dim>).
The problem is time-discretized using the implicit finite difference scheme \(\mathrm{BDF}\sigma\) (where \(\sigma = 1,2,...\) is the order of the BDF formula) as follows:
\[ \frac{\alpha_{\mathrm{BDF}\sigma} u^{n+1} - u_{\mathrm{BDF}\sigma}^n}{\Delta t} - \Delta u^{n+1} + u^{n+1} = f^{n+1}, \]
where \(\Delta t = t^{n+1}-t^{n}\) is the time step.
Boundary conditions, initial condition and source terms are provided assuming that the exact solution is:
\[ \begin{aligned} &d=2: \, u_\mathrm{ex}(x,y) &= \sin(2\pi x)\sin(2\pi y)e^{-5t}, \hspace{6mm} &(x,y) &\in \Omega=(1,1)^2&, t \in [0,T], \\ &d=3: \, u_\mathrm{ex}(x,y,z) &= \sin\left(2\pi x + \frac{\pi}{4}\right)\sin\left(2\pi y + \frac{\pi}{4}\right)\sin\left(2\pi z + \frac{\pi}{4}\right) e^{-5t}, \hspace{6mm} &(x,y,z) &\in \Omega=(1,1)^3&, t \in [0,T]. \end{aligned} \]
Finally, \(d\) is specified in the lifex configuration and \(T\) in the .prm parameter file.
Definition at line 259 of file heat_dg.hpp.
|
inline |
Constructor.
Definition at line 181 of file heat_dg.hpp.
|
overrideprivatevirtual |
Assembly of the linear system.
Implements ModelDG< basis >.
Definition at line 280 of file heat_dg.hpp.
1.9.1