Transport with dispersion implemented using discontinuous Galerkin method. More...

#include <transport_dg.hh>

Inheritance diagram for TransportDG< Model >:

Collaboration diagram for TransportDG< Model >:

[legend]

Classes
class	EqData

Public Types
enum	DGVariant { non_symmetric = -1, incomplete = 0, symmetric = 1 }

Public Member Functions
	TransportDG (Mesh &init_mesh, const Input::Record in_rec)
	Constructor. More...

void	zero_time_step () override
	Initialize solution in the zero time. More...

bool	evaluate_time_constraint (double &time_constraint)

void	update_solution () override
	Computes the solution in one time instant. More...

void	output_data ()
	Postprocesses the solution and writes to output file. More...

	~TransportDG ()
	Destructor. More...

void	initialize () override

void	calculate_cumulative_balance ()

const Vec &	get_solution (unsigned int sbi)

double **	get_concentration_matrix ()

void	calculate_concentration_matrix ()

void	update_after_reactions (bool solution_changed)

void	get_par_info (LongIdx &el_4_loc, Distribution &el_ds)

LongIdx *	get_row_4_el ()

Static Public Member Functions
static const Input::Type::Record &	get_input_type ()
	Declare input record type for the equation TransportDG. More...

static const Input::Type::Selection &	get_dg_variant_selection_input_type ()
	Input type for the DG variant selection. More...

Private Member Functions
Model::ModelEqData &	data ()

void	preallocate ()

void	assemble_mass_matrix ()
	Assembles the mass matrix. More...

template<unsigned int dim>
void	assemble_mass_matrix ()
	Assembles the mass matrix for the given dimension. More...

void	assemble_stiffness_matrix ()
	Assembles the stiffness matrix. More...

template<unsigned int dim>
void	assemble_volume_integrals ()
	Assembles the volume integrals into the stiffness matrix. More...

void	set_sources ()
	Assembles the right hand side due to volume sources. More...

template<unsigned int dim>
void	set_sources ()
	Assembles the right hand side vector due to volume sources. More...

template<unsigned int dim>
void	assemble_fluxes_boundary ()
	Assembles the fluxes on the boundary. More...

template<unsigned int dim>
void	assemble_fluxes_element_element ()
	Assembles the fluxes between elements of the same dimension. More...

template<unsigned int dim>
void	assemble_fluxes_element_side ()
	Assembles the fluxes between elements of different dimensions. More...

void	set_boundary_conditions ()
	Assembles the r.h.s. components corresponding to the Dirichlet boundary conditions. More...

template<unsigned int dim>
void	set_boundary_conditions ()
	Assembles the r.h.s. components corresponding to the Dirichlet boundary conditions for a given space dimension. More...

template<unsigned int dim>
void	calculate_velocity (const ElementAccessor< 3 > &cell, std::vector< arma::vec3 > &velocity, FEValuesBase< dim, 3 > &fv)
	Calculates the velocity field on a given `dim` dimensional cell. More...

void	set_DG_parameters_edge (const Edge &edg, const int s1, const int s2, const int K_size, const std::vector< arma::mat33 > &K1, const std::vector< arma::mat33 > &K2, const std::vector< double > &fluxes, const arma::vec3 &normal_vector, const double alpha1, const double alpha2, double &gamma, double *omega, double &transport_flux)
	Calculates the dispersivity (diffusivity) tensor from the velocity field. More...

void	set_DG_parameters_boundary (const SideIter side, const int K_size, const std::vector< arma::mat33 > &K, const double flux, const arma::vec3 &normal_vector, const double alpha, double &gamma)
	Sets up parameters of the DG method on a given boundary edge. More...

void	set_initial_condition ()
	Sets the initial condition. More...

template<unsigned int dim>
void	prepare_initial_condition ()
	Assembles the auxiliary linear system to calculate the initial solution as L^2-projection of the prescribed initial condition. More...

Private Attributes
Physical parameters
EqData	data_
	Field data for model parameters. More...

Parameters of the numerical method
FEObjects *	feo
	Finite element objects. More...

std::vector< std::vector< double > >	gamma
	Penalty parameters. More...

int	dg_variant
	DG variant ((non-)symmetric/incomplete. More...

unsigned int	dg_order
	Polynomial order of finite elements. More...

Solution of algebraic system
std::vector< Vec >	rhs
	Vector of right hand side. More...

std::vector< Mat >	stiffness_matrix
	The stiffness matrix. More...

std::vector< Mat >	mass_matrix
	The mass matrix. More...

std::vector< Vec >	mass_vec
	Mass from previous time instant (necessary when coefficients of mass matrix change in time). More...

std::vector< Vec >	ret_vec
	Auxiliary vectors for calculation of sources in balance due to retardation (e.g. sorption). More...

LinSys **	ls
	Linear algebra system for the transport equation. More...

LinSys **	ls_dt
	Linear algebra system for the time derivative (actually it is used only for handling the matrix structures). More...

double **	solution_elem_
	Element averages of solution (the array is passed to reactions in operator splitting). More...

Output to file
vector< VectorMPI >	output_vec
	Array for storing the output solution data. More...

Input::Record	input_rec
	Record with input specification. More...

Auxiliary fields used during assembly
vector< double >	mm_coef
	Mass matrix coefficients. More...

vector< vector< double > >	ret_coef
	Retardation coefficient due to sorption. More...

vector< double >	ret_sources
	Temporary values of increments due to retardation (e.g. sorption) More...

vector< double >	ret_sources_prev

vector< vector< arma::vec3 > >	ad_coef
	Advection coefficients. More...

vector< vector< arma::mat33 > >	dif_coef
	Diffusion coefficients. More...

vector< vector< vector< arma::vec3 > > >	ad_coef_edg
	Advection coefficients on edges. More...

vector< vector< vector< arma::mat33 > > >	dif_coef_edg
	Diffusion coefficients on edges. More...

Other
bool	allocation_done
	Indicates whether matrices have been preallocated. More...

Static Private Attributes
static const int	registrar
	Registrar of class to factory. More...

Detailed Description

template<class Model>
class TransportDG< Model >

Transport with dispersion implemented using discontinuous Galerkin method.

TransportDG implements the discontinuous Galerkin method for the transport and diffusion of substances. The concentration $ c_i ~[kg/m^3]$ of the i-th substance is governed by the advection-diffusion equation

$\partial_t c_i + \mathbf v\cdot\nabla c_i - \mathrm{div}(D\nabla c_i) = F \mbox{ in }\Omega^d,$

where $\mathbf v$ is the fluid velocity and $\Omega^d$ the $d$ -dimensional domain, respectively. The hydrodynamic dispersivity tensor $\mathbf D ~[m^2/s]$ is given by:

$\mathbf D = D_m\mathbf I + |\mathbf v|\left(\alpha_T\mathbf I + (\alpha_L-\alpha_T)\frac{\mathbf v\otimes\mathbf v}{|\mathbf v|^2}\right).$

The molecular dispersivity $D_m~[m^2/s]$ , as well as the longitudal and transversal dispersivity $\alpha_L,~\alpha_T~[m]$ are input parameters of the model.

For lower dimensions $d=1,2$ the advection-diffusion equation is multiplied by the fracture cross-cut $\delta^d~[m^{3-d}]$ .

The boundary $\partial\Omega^d$ is divided into three disjoint parts $\Gamma^d_D\cup\Gamma^d_N\cup\Gamma^d_F$ . We prescribe the following boundary conditions:

$\begin{eqnarray*} c_i^d &= c_{iD}^d &\mbox{ on }\Gamma^d_D \mbox{ (Dirichlet)},\\ \mathbf D^d\nabla c_i^d\cdot\mathbf n &= 0 &\mbox{ on }\Gamma^d_N \mbox{ (Neumann)}, \end{eqnarray*}$

The transfer of mass through fractures is described by the transmission conditions on $\Gamma^d_F$ :

$-\mathbf D^d\nabla c_i^d\cdot\mathbf n = \sigma(c_i^d-c_i^{d-1}) + \left\{\begin{array}{cl}0 &\mbox{ if }\mathbf v^d\cdot\mathbf n\ge 0\\\mathbf v^d\cdot\mathbf n(c_i^{d-1}-c_i^d) & \mbox{ if }\mathbf v^d\cdot\mathbf n<0\end{array}\right.,\qquad F^{d-1} = (\sigma + |\mathbf v^d\cdot\mathbf n|)(c_i^d-c_i^{d-1}).$

Here $\mathbf n$ stands for the unit outward normal vector to $\partial\Omega^d$ . The coefficient $\sigma$ determines the transfer of mass through fractures due to diffusion.

Definition at line 159 of file transport_dg.hh.

Member Enumeration Documentation

◆ DGVariant

template<class Model >

enum TransportDG::DGVariant

Enumerator
non_symmetric
incomplete
symmetric

Definition at line 179 of file transport_dg.hh.

Constructor & Destructor Documentation

◆ TransportDG()

template<class Model >

TransportDG< Model >::TransportDG	(	Mesh &	init_mesh,
		const Input::Record	in_rec
	)

Constructor.

Parameters

init_mesh	computational mesh
in_rec	input record

Definition at line 215 of file transport_dg.cc.

◆ ~TransportDG()

template<class Model >

TransportDG< Model >::~TransportDG

Destructor.

Definition at line 343 of file transport_dg.cc.

Member Function Documentation

◆ assemble_fluxes_boundary()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::assemble_fluxes_boundary

private

Assembles the fluxes on the boundary.

Definition at line 1020 of file transport_dg.cc.

◆ assemble_fluxes_element_element()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::assemble_fluxes_element_element

private

Assembles the fluxes between elements of the same dimension.

Definition at line 888 of file transport_dg.cc.

◆ assemble_fluxes_element_side()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::assemble_fluxes_element_side

private

Assembles the fluxes between elements of different dimensions.

Definition at line 1121 of file transport_dg.cc.

◆ assemble_mass_matrix() [1/2]

template<class Model >

void TransportDG< Model >::assemble_mass_matrix

private

Assembles the mass matrix.

The routine just calls templated method assemble_mass_matrix() for each space dimension.

Definition at line 663 of file transport_dg.cc.

◆ assemble_mass_matrix() [2/2]

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::assemble_mass_matrix ( )

private

Assembles the mass matrix for the given dimension.

◆ assemble_stiffness_matrix()

template<class Model >

void TransportDG< Model >::assemble_stiffness_matrix

private

Assembles the stiffness matrix.

This routine just calls assemble_volume_integrals(), assemble_fluxes_boundary(), assemble_fluxes_element_element() and assemble_fluxes_element_side() for each space dimension.

Definition at line 731 of file transport_dg.cc.

◆ assemble_volume_integrals()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::assemble_volume_integrals

private

Assembles the volume integrals into the stiffness matrix.

Definition at line 764 of file transport_dg.cc.

◆ calculate_concentration_matrix()

template<class Model >

void TransportDG< Model >::calculate_concentration_matrix

Definition at line 578 of file transport_dg.cc.

◆ calculate_cumulative_balance()

template<class Model >

void TransportDG< Model >::calculate_cumulative_balance

Definition at line 644 of file transport_dg.cc.

◆ calculate_velocity()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::calculate_velocity	(	const ElementAccessor< 3 > &	cell,
		std::vector< arma::vec3 > &	velocity,
		FEValuesBase< dim, 3 > &	fv
	)

private

Calculates the velocity field on a given dim dimensional cell.

Parameters

cell	The cell.
velocity	The computed velocity field (at quadrature points).
fv	The FEValues class providing the quadrature points and the shape functions for velocity.

Definition at line 1409 of file transport_dg.cc.

◆ data()

template<class Model >

Model::ModelEqData& TransportDG< Model >::data ( )

inlineprivate

Definition at line 255 of file transport_dg.hh.

◆ evaluate_time_constraint()

template<class Model >

bool TransportDG< Model >::evaluate_time_constraint ( double & time_constraint )

inline

Definition at line 212 of file transport_dg.hh.

◆ get_concentration_matrix()

template<class Model >

double** TransportDG< Model >::get_concentration_matrix ( )

inline

Definition at line 237 of file transport_dg.hh.

◆ get_dg_variant_selection_input_type()

template<class Model >

const Selection & TransportDG< Model >::get_dg_variant_selection_input_type

static

Input type for the DG variant selection.

Definition at line 53 of file transport_dg.cc.

◆ get_input_type()

template<class Model >

const Record & TransportDG< Model >::get_input_type

static

Declare input record type for the equation TransportDG.

Definition at line 79 of file transport_dg.cc.

◆ get_par_info()

template<class Model >

void TransportDG< Model >::get_par_info	(	LongIdx *&	el_4_loc,
		Distribution *&	el_ds
	)

Definition at line 1660 of file transport_dg.cc.

◆ get_row_4_el()

template<class Model >

LongIdx * TransportDG< Model >::get_row_4_el

Definition at line 1712 of file transport_dg.cc.

◆ get_solution()

template<class Model >

const Vec& TransportDG< Model >::get_solution ( unsigned int sbi )

inline

Definition at line 234 of file transport_dg.hh.

◆ initialize()

template<class Model >

void TransportDG< Model >::initialize

override

Definition at line 249 of file transport_dg.cc.

◆ output_data()

template<class Model >

void TransportDG< Model >::output_data

Postprocesses the solution and writes to output file.

Definition at line 619 of file transport_dg.cc.

◆ preallocate()

template<class Model >

void TransportDG< Model >::preallocate

private

Definition at line 416 of file transport_dg.cc.

◆ prepare_initial_condition()

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::prepare_initial_condition

private

Assembles the auxiliary linear system to calculate the initial solution as L^2-projection of the prescribed initial condition.

Definition at line 1610 of file transport_dg.cc.

◆ set_boundary_conditions() [1/2]

template<class Model >

void TransportDG< Model >::set_boundary_conditions

private

Assembles the r.h.s. components corresponding to the Dirichlet boundary conditions.

The routine just calls templated method set_boundary_condition() for each space dimension.

Definition at line 1242 of file transport_dg.cc.

◆ set_boundary_conditions() [2/2]

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::set_boundary_conditions ( )

private

Assembles the r.h.s. components corresponding to the Dirichlet boundary conditions for a given space dimension.

◆ set_DG_parameters_boundary()

template<class Model >

void TransportDG< Model >::set_DG_parameters_boundary	(	const SideIter	side,
		const int	K_size,
		const std::vector< arma::mat33 > &	K,
		const double	flux,
		const arma::vec3 &	normal_vector,
		const double	alpha,
		double &	gamma
	)

private

Sets up parameters of the DG method on a given boundary edge.

Assumption is that the edge consists of only 1 side.

Parameters

side	The boundary side.
K_size	Size of vector of tensors K.
K	Dispersivity tensor.
ad_vector	Advection vector.
normal_vector	Normal vector (assumed constant along the edge).
alpha	Penalty parameter that influences the continuity of the solution (large value=more continuity).
gamma	Computed penalty parameters.

Definition at line 1550 of file transport_dg.cc.

◆ set_DG_parameters_edge()

template<class Model >

void TransportDG< Model >::set_DG_parameters_edge	(	const Edge &	edg,
		const int	s1,
		const int	s2,
		const int	K_size,
		const std::vector< arma::mat33 > &	K1,
		const std::vector< arma::mat33 > &	K2,
		const std::vector< double > &	fluxes,
		const arma::vec3 &	normal_vector,
		const double	alpha1,
		const double	alpha2,
		double &	gamma,
		double *	omega,
		double &	transport_flux
	)

private

Calculates the dispersivity (diffusivity) tensor from the velocity field.

Parameters

K	The computed dispersivity tensor.
velocity	The velocity field (at quadrature points).
Dm	Molecular diffusivities.
alphaL	Longitudal dispersivities.
alphaT	Transversal dispersivities.
porosity	Porosities.
cross_cut	Cross-cuts of higher dimension.

Sets up some parameters of the DG method for two sides of an edge.

Parameters

edg	The edge.
s1	Side 1.
s2	Side 2.
K_size	Size of vector of tensors K.
K1	Dispersivity tensors on side s1 (in quadrature points).
K2	Dispersivity tensors on side s2 (in quadrature points).
normal_vector	Normal vector to side 0 of the neighbour (assumed constant along the side).
alpha1,alpha2	Penalty parameter that influences the continuity of the solution (large value=more continuity).
gamma	Computed penalty parameters.
omega	Computed weights.
transport_flux	Computed flux from side s1 to side s2.

Definition at line 1443 of file transport_dg.cc.

◆ set_initial_condition()

template<class Model >

void TransportDG< Model >::set_initial_condition

private

Sets the initial condition.

Definition at line 1585 of file transport_dg.cc.

◆ set_sources() [1/2]

template<class Model >

void TransportDG< Model >::set_sources

private

Assembles the right hand side due to volume sources.

This method just calls set_sources() for each space dimension.

Definition at line 817 of file transport_dg.cc.

◆ set_sources() [2/2]

template<class Model >

template<unsigned int dim>

void TransportDG< Model >::set_sources ( )

private

Assembles the right hand side vector due to volume sources.

◆ update_after_reactions()

template<class Model >

void TransportDG< Model >::update_after_reactions ( bool solution_changed )

Definition at line 1668 of file transport_dg.cc.

◆ update_solution()

template<class Model >

void TransportDG< Model >::update_solution

override

Computes the solution in one time instant.

Definition at line 447 of file transport_dg.cc.

◆ zero_time_step()

template<class Model >

void TransportDG< Model >::zero_time_step

override

Initialize solution in the zero time.

Definition at line 383 of file transport_dg.cc.

Member Data Documentation

◆ ad_coef

template<class Model >

vector<vector<arma::vec3> > TransportDG< Model >::ad_coef

private

Advection coefficients.

Definition at line 511 of file transport_dg.hh.

◆ ad_coef_edg

template<class Model >

vector<vector<vector<arma::vec3> > > TransportDG< Model >::ad_coef_edg

private

Advection coefficients on edges.

Definition at line 515 of file transport_dg.hh.

◆ allocation_done

template<class Model >

bool TransportDG< Model >::allocation_done

private

Indicates whether matrices have been preallocated.

Definition at line 528 of file transport_dg.hh.

◆ data_

template<class Model >

EqData TransportDG< Model >::data_

private

Field data for model parameters.

Definition at line 431 of file transport_dg.hh.

◆ dg_order

template<class Model >

unsigned int TransportDG< Model >::dg_order

private

Polynomial order of finite elements.

Definition at line 449 of file transport_dg.hh.

◆ dg_variant

template<class Model >

int TransportDG< Model >::dg_variant

private

DG variant ((non-)symmetric/incomplete.

Definition at line 446 of file transport_dg.hh.

◆ dif_coef

template<class Model >

vector<vector<arma::mat33> > TransportDG< Model >::dif_coef

private

Diffusion coefficients.

Definition at line 513 of file transport_dg.hh.

◆ dif_coef_edg

template<class Model >

vector<vector<vector<arma::mat33> > > TransportDG< Model >::dif_coef_edg

private

Diffusion coefficients on edges.

Definition at line 517 of file transport_dg.hh.

◆ feo

template<class Model >

FEObjects* TransportDG< Model >::feo

private

Finite element objects.

Definition at line 440 of file transport_dg.hh.

◆ gamma

template<class Model >

std::vector<std::vector<double> > TransportDG< Model >::gamma

private

Penalty parameters.

Definition at line 443 of file transport_dg.hh.

◆ input_rec

template<class Model >

Input::Record TransportDG< Model >::input_rec

private

Record with input specification.

Definition at line 495 of file transport_dg.hh.

◆ ls

template<class Model >

LinSys** TransportDG< Model >::ls

private

Linear algebra system for the transport equation.

Definition at line 474 of file transport_dg.hh.

◆ ls_dt

template<class Model >

LinSys** TransportDG< Model >::ls_dt

private

Linear algebra system for the time derivative (actually it is used only for handling the matrix structures).

Definition at line 477 of file transport_dg.hh.

◆ mass_matrix

template<class Model >

std::vector<Mat> TransportDG< Model >::mass_matrix

private

The mass matrix.

Definition at line 465 of file transport_dg.hh.

◆ mass_vec

template<class Model >

std::vector<Vec> TransportDG< Model >::mass_vec

private

Mass from previous time instant (necessary when coefficients of mass matrix change in time).

Definition at line 468 of file transport_dg.hh.

◆ mm_coef

template<class Model >

vector<double> TransportDG< Model >::mm_coef

private

Mass matrix coefficients.

Definition at line 505 of file transport_dg.hh.

◆ output_vec

template<class Model >

vector<VectorMPI> TransportDG< Model >::output_vec

private

Array for storing the output solution data.

Vector of solution data.

Definition at line 492 of file transport_dg.hh.

◆ registrar

template<class Model >

const int TransportDG< Model >::registrar

staticprivate

Initial value:

=
        Input::register_class< TransportDG<Model>, Mesh &, const Input::Record>(std::string(Model::ModelEqData::name()) + "_DG") +
        TransportDG<Model>::get_input_type().size()

Registrar of class to factory.

Definition at line 253 of file transport_dg.hh.

◆ ret_coef

template<class Model >

vector<vector<double> > TransportDG< Model >::ret_coef

private

Retardation coefficient due to sorption.

Definition at line 507 of file transport_dg.hh.

◆ ret_sources

template<class Model >

vector<double> TransportDG< Model >::ret_sources

private

Temporary values of increments due to retardation (e.g. sorption)

Definition at line 509 of file transport_dg.hh.

◆ ret_sources_prev

template<class Model >

vector<double> TransportDG< Model >::ret_sources_prev

private

Definition at line 509 of file transport_dg.hh.

◆ ret_vec

template<class Model >

std::vector<Vec> TransportDG< Model >::ret_vec

private

Auxiliary vectors for calculation of sources in balance due to retardation (e.g. sorption).

Definition at line 471 of file transport_dg.hh.

◆ rhs

template<class Model >

std::vector<Vec> TransportDG< Model >::rhs

private

Vector of right hand side.

Definition at line 459 of file transport_dg.hh.

◆ solution_elem_

template<class Model >

double** TransportDG< Model >::solution_elem_

private

Element averages of solution (the array is passed to reactions in operator splitting).

Definition at line 480 of file transport_dg.hh.

◆ stiffness_matrix

template<class Model >

std::vector<Mat> TransportDG< Model >::stiffness_matrix

private

The stiffness matrix.

Definition at line 462 of file transport_dg.hh.

The documentation for this class was generated from the following files:

/opt/flow123d/flow123d/src/transport/transport_dg.hh
/opt/flow123d/flow123d/src/transport/transport_dg.cc

Classes

Public Types

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

template<class Model> class TransportDG< Model >

Member Enumeration Documentation

◆ DGVariant

Constructor & Destructor Documentation

◆ TransportDG()

◆ ~TransportDG()

Member Function Documentation

◆ assemble_fluxes_boundary()

◆ assemble_fluxes_element_element()

◆ assemble_fluxes_element_side()

◆ assemble_mass_matrix() [1/2]

◆ assemble_mass_matrix() [2/2]

◆ assemble_stiffness_matrix()

◆ assemble_volume_integrals()

◆ calculate_concentration_matrix()

◆ calculate_cumulative_balance()

◆ calculate_velocity()

◆ data()

◆ evaluate_time_constraint()

◆ get_concentration_matrix()

◆ get_dg_variant_selection_input_type()

◆ get_input_type()

◆ get_par_info()

◆ get_row_4_el()

◆ get_solution()

◆ initialize()

◆ output_data()

◆ preallocate()

◆ prepare_initial_condition()

◆ set_boundary_conditions() [1/2]

◆ set_boundary_conditions() [2/2]

◆ set_DG_parameters_boundary()

◆ set_DG_parameters_edge()

◆ set_initial_condition()

◆ set_sources() [1/2]

◆ set_sources() [2/2]

◆ update_after_reactions()

◆ update_solution()

◆ zero_time_step()

Member Data Documentation

◆ ad_coef

◆ ad_coef_edg

◆ allocation_done

◆ data_

◆ dg_order

◆ dg_variant

◆ dif_coef

◆ dif_coef_edg

◆ feo

◆ gamma

◆ input_rec

◆ ls

◆ ls_dt

◆ mass_matrix

◆ mass_vec

◆ mm_coef

◆ output_vec

◆ registrar

◆ ret_coef

◆ ret_sources

◆ ret_sources_prev

◆ ret_vec

◆ rhs

◆ solution_elem_

◆ stiffness_matrix

template<class Model>
class TransportDG< Model >