implements a mixed-hybrid FE Darcy solver for transient problems More...

#include <DarcySolverTransient.hpp>

Inheritance diagram for DarcySolverTransient< MeshType >:

Collaboration diagram for DarcySolverTransient< MeshType >:

Public Types
typedef MeshType	mesh_Type
	Typedef for mesh template. More...

typedef DarcySolverTransient< mesh_Type >	darcySolverTransient_Type
	Self typedef. More...

typedef DarcySolverLinear< mesh_Type >	darcySolverLinear_Type
	Darcy solver class. More...

typedef darcySolverLinear_Type::data_Type	data_Type
	Typedef for the data type. More...

typedef darcySolverLinear_Type::dataPtr_Type	dataPtr_Type
	Shared pointer for the data type. More...

typedef darcySolverLinear_Type::scalarFctPtr_Type	scalarFctPtr_Type
	Shared pointer to a scalar value function. More...

typedef darcySolverLinear_Type::vectorPtr_Type	vectorPtr_Type
	Shared pointer to a distributed vector. More...

typedef darcySolverLinear_Type::vector_Type	vector_Type
	Distributed vector. More...

typedef darcySolverLinear_Type::scalarFieldPtr_Type	scalarFieldPtr_Type
	Shared pointer to a scalar field. More...

typedef darcySolverLinear_Type::scalarField_Type	scalarField_Type
	Scalar field. More...

typedef std::vector< MatrixElemental >	matrixElementalContainer_Type
	Container of matrix elemental. More...

typedef TimeAdvanceBDF< vector_Type >	timeAdvance_Type
	Time advance scheme. More...

typedef std::shared_ptr< timeAdvance_Type >	timeAdvancePtr_Type
	Shared pointer to a time advance scheme. More...

Constructors & destructor
	DarcySolverTransient ()
	Constructor for the class. More...

virtual	~DarcySolverTransient ()
	Virtual destructor. More...

methods
virtual void	setup ()
	Set up the linear solver and the preconditioner for the linear system. More...

virtual void	solve ()
	Solve the problem. More...

Update and set methods
void	setInitialPrimal (const scalarFctPtr_Type &primalInitialFct)
	Initialize primal solution. More...

void	setMass (const scalarFctPtr_Type &massFct)
	Set mass matrix. More...

Protected methods
virtual void	localMatrixComputation (const UInt &iElem, MatrixElemental &elmatMix, MatrixElemental &elmatReactionTerm)
	Compute element matrices. More...

virtual void	localVectorComputation (const UInt &iElem, VectorElemental &elvecMix)
	Computes local vectors. More...

virtual void	postComputePrimalAndDual ()
	Do some computation after the calculation of the primal and dual variable. More...

void	setupTime ()
	Setup the time data. More...

Time advance stuff
timeAdvancePtr_Type	M_timeAdvance
	Time advance. More...

vectorPtr_Type	M_rhsTimeAdvance
	Right hand side coming from the time advance scheme. More...

matrixElementalContainer_Type	M_localMassMatrix
	Local mass matrices. More...

Private Constructors
	DarcySolverTransient (const darcySolverTransient_Type &)
	Inhibited copy constructor. More...

Private Operators
darcySolverTransient_Type &	operator= (const darcySolverTransient_Type &)
	Inhibited assign operator. More...

Data of the problem
scalarFctPtr_Type	M_primalFieldInitialFct
	Initial time primal variable. More...

scalarFctPtr_Type	M_massFct
	Mass function, it does not depend on time. More...

Algebraic stuff
bool	M_reusePrec
	Boolean that indicates if the preconditioner is re-used or not. More...

bool	M_updated
	Boolean that indicates if the matrix is updated for the current iteration. More...

UInt	M_maxIterSolver
	Interger storing the max number of solver iteration with preconditioner recomputing. More...

bool	M_recomputeMatrix
	Boolean that indicates if the matrix is recomputed for the current iteration. More...

Additional Inherited Members
Public Types inherited from DarcySolverLinear< MeshType >
typedef MeshType	mesh_Type
	Typedef for mesh template. More...

typedef LinearSolver	solver_Type
	Typedef for solver template. More...

typedef DarcySolverLinear< mesh_Type >	darcySolver_Type
	Self typedef. More...

typedef DarcyData< mesh_Type >	data_Type
	Typedef for the data type. More...

typedef std::shared_ptr< data_Type >	dataPtr_Type
	Shared pointer for the data type. More...

typedef BCHandler	bcHandler_Type
	Boundary condition handler. More...

typedef std::shared_ptr< bcHandler_Type >	bcHandlerPtr_Type
	Shared pointer to a boundary condition handler. More...

typedef std::shared_ptr< Epetra_Comm >	commPtr_Type
	Shared pointer to a MPI communicator. More...

typedef MapEpetra	map_Type
	Map type. More...

typedef std::shared_ptr< Displayer >	displayerPtr_Type
	Shared pointer to a displayer. More...

typedef FESpace< mesh_Type, map_Type >	fESpace_Type
	Finite element space. More...

typedef std::shared_ptr< fESpace_Type >	fESpacePtr_Type
	Shared pointer to a finite element space. More...

typedef FEScalarField< mesh_Type, map_Type >	scalarField_Type
	Scalar field. More...

typedef std::shared_ptr< scalarField_Type >	scalarFieldPtr_Type
	Shared pointer to a scalar field. More...

typedef FEVectorField< mesh_Type, map_Type >	vectorField_Type
	Vector field. More...

typedef std::shared_ptr< vectorField_Type >	vectorFieldPtr_Type
	Shared pointer to a scalar field. More...

typedef FEFunction< mesh_Type, map_Type, Real >	scalarFct_Type
	Scalar value function. More...

typedef std::shared_ptr< scalarFct_Type >	scalarFctPtr_Type
	Shared pointer to a scalar value function. More...

typedef FEFunction< mesh_Type, map_Type, Vector >	vectorFct_Type
	Vector value function. More...

typedef std::shared_ptr< vectorFct_Type >	vectorFctPtr_Type
	Shared pointer to a vector value function. More...

typedef FEFunction< mesh_Type, map_Type, Matrix >	matrixFct_Type
	Matrix value funcion. More...

typedef std::shared_ptr< matrixFct_Type >	matrixFctPtr_Type
	Shared pointer to a matrix value function. More...

typedef solver_Type::matrix_Type	matrix_Type
	Sparse and distributed matrix. More...

typedef solver_Type::matrixPtr_Type	matrixPtr_Type
	Shared pointer to a sparse and distributed matrix. More...

typedef solver_Type::vector_Type	vector_Type
	Distributed vector. More...

typedef solver_Type::vectorPtr_Type	vectorPtr_Type
	Shared pointer to a distributed vector. More...

typedef solver_Type::preconditionerPtr_Type	preconditionerPtr_Type
	Shared pointer to the preconditioner. More...

Public Member Functions inherited from DarcySolverLinear< MeshType >
	DarcySolverLinear ()
	Constructor for the class. More...

virtual	~DarcySolverLinear ()
	Virtual destructor. More...

void	buildSystem ()
	Build the global hybrid system, the right hand and apply the boundary conditions. More...

void	solveLinearSystem ()
	Solve the global hybrid system. More...

void	computePrimalAndDual ()
	Compute primal and dual variables from the hybrid variable as a post process. More...

void	setData (dataPtr_Type &data)
	Set the data. More...

void	setBoundaryConditions (bcHandlerPtr_Type &bcHandler)
	Set the boundary conditions. More...

void	setScalarSource (const scalarFctPtr_Type &scalarSourceFct)
	Set scalar source term. More...

void	setVectorSource (const vectorFctPtr_Type &vectorSourceFct)
	Set vector source term. More...

virtual void	setInversePermeability (const matrixFctPtr_Type &invPermFct)
	Set inverse diffusion tensor. More...

void	setReactionTerm (const scalarFctPtr_Type &reactionTermFct)
	Set the coefficient for the reaction term. More...

void	setHybridField (const scalarFieldPtr_Type &hybridField)
	Set the hybrid field vector. More...

void	setPrimalField (const scalarFieldPtr_Type &primalField)
	Set the primal field vector. More...

void	setDualField (const scalarFieldPtr_Type &dualField)
	Set the dual field vector. More...

void	setFields (const scalarFieldPtr_Type &dualField, const scalarFieldPtr_Type &primalField, const scalarFieldPtr_Type &hybridField)
	Set the fields. More...

void	setDisplayer (const displayerPtr_Type &displayer)
	Set the displayer and, implicitly, the communicator. More...

void	setCommunicator (const commPtr_Type &comm)
	Set the communicator and, implicitly, the displayer. More...

const scalarFieldPtr_Type &	hybridFieldPtr () const
	Returns pointer to the hybrid solution field. More...

scalarFieldPtr_Type &	hybridFieldPtr ()
	Returns pointer to the hybrid solution field. More...

const scalarFieldPtr_Type &	primalFieldPtr () const
	Returns pointer to the primal solution field. More...

scalarFieldPtr_Type &	primalFieldPtr ()
	Returns pointer to the primal solution field. More...

const scalarFieldPtr_Type &	dualFieldPtr () const
	Returns pointer to the dual solution field. More...

scalarFieldPtr_Type &	dualFieldPtr ()
	Returns pointer to the dual solution field. More...

const vectorPtr_Type &	residualPtr () const

vectorPtr_Type &	residualPtr ()

const bcHandlerPtr_Type &	boundaryConditionHandlerPtr () const
	Returns boundary conditions handler. More...

bcHandlerPtr_Type &	boundaryConditionHandlerPtr ()
	Returns boundary conditions handler. More...

const map_Type &	getMap () const
	Returns Epetra local map. More...

const commPtr_Type &	getCommPtr () const
	Returns Epetra communicator. More...

Protected Member Functions inherited from DarcySolverLinear< MeshType >
	DarcySolverLinear (const darcySolver_Type &)
	Inhibited copy constructor. More...

darcySolver_Type &	operator= (const darcySolver_Type &)
	Inhibited assign operator. More...

virtual void	preLoopElementsComputation ()
	Perform all the operations before doing the loop on volume elements. More...

void	computeConstantMatrices (MatrixElemental &elmatMix)
	Pre-computes local (element independant) matrices. More...

void	staticCondensation (MatrixElemental &localMatrixHybrid, VectorElemental &localVectorHybrid, MatrixElemental &elmatMix, MatrixElemental &elmatReactionTerm, VectorElemental &elvecMix)
	Performs static condensation. More...

void	localComputePrimalAndDual (VectorElemental &localSolution, MatrixElemental &elmatMix, MatrixElemental &elmatReactionTerm, VectorElemental &elvecMix)
	Compute locally, as a post process, the primal and dual variable given the hybrid. More...

virtual void	resetVariables ()
	Update all problem variables. More...

void	applyBoundaryConditions ()
	Apply the boundary condition. More...

template<typename MatrixType >
void	symmetrizeMatrix (Int N, MatrixType &A)
	Make matrix symmetric. More...

Protected Attributes inherited from DarcySolverLinear< MeshType >
displayerPtr_Type	M_displayer
	Displayer for parallel cout. More...

dataPtr_Type	M_data
	Data for Darcy solvers. More...

scalarFctPtr_Type	M_scalarSourceFct
	Source function. More...

vectorFctPtr_Type	M_vectorSourceFct
	Vector source function. More...

scalarFctPtr_Type	M_reactionTermFct
	Reaction term function. More...

matrixFctPtr_Type	M_inversePermeabilityFct
	Inverse of the permeability tensor. More...

bcHandlerPtr_Type	M_boundaryConditionHandler
	Bondary conditions handler. More...

scalarFieldPtr_Type	M_primalField
	Primal solution. More...

scalarFieldPtr_Type	M_dualField
	Dual solution. More...

scalarFieldPtr_Type	M_hybridField
	Hybrid solution. More...

matrixPtr_Type	M_matrHybrid
	Hybrid matrix. More...

vectorPtr_Type	M_rhs
	Right hand side. More...

vectorPtr_Type	M_residual
	Residual. More...

solver_Type	M_linearSolver
	Linear solver. More...

preconditionerPtr_Type	M_prec
	Epetra preconditioner for the linear system. More...

Detailed Description

template<typename MeshType>
class LifeV::DarcySolverTransient< MeshType >

implements a mixed-hybrid FE Darcy solver for transient problems

Author: A. Fumagalli aless.nosp@m.io.f.nosp@m.umaga.nosp@m.lli@.nosp@m.mail..nosp@m.poli.nosp@m.mi.it

This class implements a transient Darcy solver in the finite time interval $ [0, T] $ .
The classical time dependent Darcy formulation is a couple of differential equations of first order with the unknowns $p \in C^1(0, T; C^1 (\Omega ))$ , being the pressure or the primal unknown, and $\sigma \in C^0(0, T; (C^1( \Omega ) )^n)$ , being the Darcy velocity or the flux or the dual unknown, such that

$\left\{ \begin{array}{l l l} \Lambda^{-1}(t) \sigma + \nabla p = f_v(t) & \mathrm{in} & \Omega \times [0, T]\,, \vspace{0.2cm} \\ \displaystyle \phi \frac{\partial p}{\partial t} + \nabla \cdot \sigma + \pi(t) p - f(t) = 0 & \mathrm{in} & \Omega \times [0, T]\,, \vspace{0.2cm} \\ p = g_D(t) & \mathrm{on} & \Gamma_D \times [0, T]\,, \vspace{0.2cm} \\ \sigma \cdot n + h(t) p = g_R(t) & \mathrm{on} & \Gamma_R \times [0, T]\,. \vspace{0.2cm} \\ p(0) = p_0 & \mathrm{in} & \Omega \end{array} \right.$

The data in the system are:

the primal variable at initial time;
$\phi$ the mass term, which does not depend on time;
$\Lambda(t)$ the permeability tensor;
the scalar source term;
the vector source term;
$\pi(t)$ the reaction coefficient;
$\Gamma_D$ the subset of the boundary of $\Omega$ with Dirichlet boundary conditions;
Dirichlet boundary condition data;
$\Gamma_R$ that is the part of the boundary of $\Omega$ with Robin, or Neumann, boundary conditions;
and Robin boundary condition data;

We suppose that $\partial \Omega = \Gamma_D \cup \Gamma_R$ and $\Gamma_D \cap \Gamma_R = \emptyset$ .
Using the hybridization procedure, and introducing a new variable, we may split the problem from the entire domain to problems in each elements of the triangulation $\mathcal{T}_h$ , then we may write the weak formulation for the dual problem. The new variable is the hybrid unknown $\lambda$ , e.g. "the trace" of pressure of the primal unknown, it is the Lagrange multipliers forcing the continuity of the flux across each faces or edges in the triangulation. We introduce the following functional spaces, the first is the space of the primal variable, the third for the dual variable, the fourth for the hybrid variable.

$\begin{array}{l} V = L^2 (\Omega ) \,,\vspace{0.2cm} \\ H(div, K) = \displaystyle \left\{ \tau \in ( L^2(K))^n : \nabla \cdot \tau \in L^2(K)\right\}\,, \vspace{0.2cm}\\ Z = \displaystyle \left\{ \tau \in L^2(\Omega) : \tau\vert_K \in H(div, K) \, \forall K \in \mathcal{T}_h \right\}\,, \vspace{0.2cm}\\ \Lambda = \displaystyle \left\{ \lambda \in \prod_{K \in \mathcal{T}_h} H^{1/2} (\partial K): \lambda_K = \lambda_{K'} \,\, \mathrm{on} \,\, e_{K-K'} \, \forall K \in \mathcal{T}_h,\, \lambda = g_D \,\, \mathrm{on} \,\, \Gamma_D \right\}\,. \end{array}$

Introducing the following bilinear forms and functionals at fixed time $t \in [0, T]$

$\begin{array}{l l} a(\sigma(t), \tau) = \displaystyle \sum_{K \in \mathcal{T}_h}\int_K \Lambda^{-1}(t) \sigma(t) \cdot \tau \,, & b(p(t), \tau) = \displaystyle -\sum_{K \in \mathcal{T}_h} \int_K p(t) \nabla \cdot \tau\,, \vspace{0.2cm}\\ c(\lambda(t), \tau) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_{\partial K} \lambda(t) \tau \cdot n\,,& d(p(t), v) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_{\partial K} \pi(t) p(t) v \,,\vspace{0.2cm}\\ h(\lambda(t), \mu) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_{\partial K \cap \Gamma_R} h(t) \mu \lambda(t) \,,& m(p(t), v) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_K \phi p(t) v \,,\vspace{0.2cm}\\ F(v) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_K f(t) v\,,& F_v(\tau) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_K f_v(t) \tau \,, \vspace{0.2cm}\\ G(\mu) = \displaystyle \sum_{K \in \mathcal{T}_h} \int_{\partial K \cap \Gamma_R} g(t) \mu\,, \end{array}$

we obtain the Darcy problem in the weak form: find $(\sigma(t), \, p(t), \, \lambda(t)) \in Z \times V \times \Lambda$ such that

$\left\{ \begin{array}{l l} a(\sigma(t), \tau) + b(p(t), \tau) + c(\lambda(t), \tau) = F_v(\tau) \,, & \forall \tau \in Z \,,\vspace{0.2cm}\\ \displaystyle -\frac{\partial }{\partial t} m(p(t), v) + b(v, \sigma(t)) - d(p(t), v) = -F(v)\,, & \forall v \in V \,,\vspace{0.2cm}\\ c(\mu, \sigma(t)) + h(\lambda(t), \mu) = G(\mu) \,, & \forall \mu \in \Lambda\,. \end{array} \right.$

At semi-discrete level. i.e. only space discretization is performed, we introduce the polynomial space, of degree $ r $ , that approximate the finite dimensional spaces introduced above $V_h \subset V$ , $Z_h \subset Z$ and $\Lambda_h \subset \Lambda$

$\begin{array}{l} V_h = \displaystyle \left\{ v_h \in V: v_h|_K \in P_r (K)\, \forall K \in \mathcal{T}_h \right\}\,, \vspace{0.2cm}\\ Z_h = \displaystyle \left\{ \tau_h \in Z: \tau_h|_K \in RT_r(K) \, \forall K \in \mathcal{T}_h \right\} \,, \vspace{0.2cm} \\ \Lambda_h = \displaystyle \left\{ \lambda_h \in \Lambda: \lambda_h|_{\partial K} \in R_r( \partial K ) \, \forall K \in \mathcal{T}_h\right\}\,, \end{array}$

where $ P_r(K) $ is the space of polynomial of degree $ r $ in the element $ K $ , $ RT_r(K) $ is the space of polynomial of Raviart-Thomas of degrees $ r $ in the element $ K $ and $R_r(\partial K)$ is the space of polynomial of degree $ r $ definite on each of the boundary of the element $ K $ and discontinuous from one edge to the other.
The finite dimensional problem is: find $(\sigma_h(t),\, p_h(t), \, \lambda_h(t)) \in Z_h \times V_h \times \Lambda_h$ such that

$\left\{ \begin{array}{l l} a(\sigma_h(t), \tau_h) + b(p_h(t), \tau_h) + c(\lambda_h(t), \tau_h) = F_v(\tau_h) \,, & \forall \tau_h \in Z_h \,,\vspace{0.2cm}\\ \displaystyle -\frac{\partial }{\partial t} m(p_h(t),v_h) + b(v_h, \sigma_h(t)) - d(p_h(t), v_h) = -F(v_h)\,, & \forall v_h \in V_h \,,\vspace{0.2cm}\\ c(\mu_h, \sigma_h(t)) + h(\lambda_h(t), \mu_h) = G(\mu_h) \,, & \forall \mu_h \in \Lambda_h\,. \end{array} \right.$

To obatin a fully discrete problem we discretize the time derivative via BDF (backward differentiation formulae) schemes of order $ m $ , with $\Delta t$ as time step. Choosing

$p_h^0 = \prod_{V_h} p_0$

we obtain the following system for each $n = 0, \ldots, N$ , with $N = \frac{T}{\Delta t}$

$\left\{ \begin{array}{l l} a(\sigma_h^{n+1}, \tau_h) + b(p_h^{n+1}, \tau_h) + c(\lambda_h^{n+1}, \tau_h) = F_v(\tau_h)\,, & \forall \tau_h \in Z_h \,,\vspace{0.2cm}\\ \displaystyle -\frac{\alpha_0}{\Delta t} m(p_h^{n+1},v_h) + b(v_h, \sigma_h^{n+1}) - d(p_h^{n+1}, v_h)= -F(v_h) - \frac{1}{\Delta t} m\left( \sum_{i=1}^m \alpha_i p_h^{n+1-i}, v_h \right) \,, & \forall v_h \in V_h \,,\vspace{0.2cm}\\ c(\mu_h, \sigma_h^{n+1}) + h(\lambda_h^{n+1}, \mu_h) = G(\mu_h) \,, & \forall \mu_h \in \Lambda_h\,. \end{array} \right.$

To solve the problem we use the static condensation procedure, i.e. the unknowns in the discrete weak system are not independent and $ p_K $ , $\sigma_K$ may be written in function of $\lambda_K$ alone. We introduce the following local matrices

$\begin{array}{l l l} \left[ A \right]_{ij} = \displaystyle \int_K \Lambda^{-1}( (n+1) \Delta t) \psi_j \cdot \psi_i \,, & \left[ B \right]_{ij} = \displaystyle - \int_K \phi_j \nabla \cdot \psi_i \,, & \left[ C \right]_{ij} = \displaystyle \int_{\partial K} \xi_i \psi_j \cdot n \,, \vspace{0.2cm} \\ \left[ D \right]_{ij} = \displaystyle \int_K \pi(t) \phi_j \phi_i\,, & \left[ H \right]_{ij} = \displaystyle \int_{\partial K \cap \Gamma_R} h( (n+1) \Delta t) \xi_i \xi_j\,, & \left[ M \right]_{ij} = \displaystyle \int_{K} \phi \phi_j \phi_i \,, \vspace{0.2cm} \\ \left[ F \right]_{j} = \displaystyle \int_K f( (n+1) \Delta t) \phi_j\,,& \left[ F_v \right]_{j}= \displaystyle \int_K f_v( (n+1) \Delta t) \cdot \psi_j\,, & \left[ G \right]_{j} = \displaystyle \int_{\partial K \cap \Gamma_R } g( (n+1) \Delta t) \xi_j\,, \end{array}$

where we avoid to write the dependence on the triangle $ K $ and on the current time step $ n+1 $ in all the matrices and vectors.
bre local matrix formulation of the finite dimensional problem is

$\left\{ \begin{array}{l} A \sigma_K^{n+1} + B p_K^{n+1} + C \lambda_K^{n+1} = F_v\,, \vspace{0.2cm} \\ \displaystyle -\frac{\alpha_0}{\Delta t} M p_K^{n+1} + B^T \sigma_K^{n+1} - D p_K^{n+1} = -F - \frac{1}{\Delta t} M \sum_{i=1}^m \alpha_i p_K^{n+1-i} \,,\vspace{0.2cm}\\ C^T \sigma_K^{n+1} + H \lambda_K^{n+1} = G\,. \end{array} \right.$

Or alternatively

$\begin{array}{l l l} \left[ \begin{array}{c c c} A & B & C \vspace{0.2cm} \\ B^T & \displaystyle -\frac{\alpha_0}{\Delta t} M - D & 0 \vspace{0.2cm} \\ C^T & 0 & H \end{array} \right] \, \cdot & \left[ \begin{array}{c} \sigma_K^{n+1} \vspace{0.2cm}\\ p_K^{n+1} \vspace{0.2cm}\\ \lambda_K^{n+1} \end{array} \right] \, & = \left[ \begin{array}{c} F_v \vspace{0.2cm}\\ \displaystyle - F - \frac{1}{\Delta t} M \sum_{i=1}^m \alpha_i p_K^{n+1-i} \vspace{0.2cm}\\ G \end{array} \right]\,. \end{array}$

Introducing the local hybrid matrix and local hybrid right hand side

$\begin{array}{l} \displaystyle L_K = -C^T A^{-1} C + C^T A^{-1} B \left( \frac{\alpha_0}{\Delta t} M + B^T A^{-1} B + D \right)^{-1} B^T A^{-1} C + H \,, \vspace{0.2cm} \\ \displaystyle r_K = G + C^T A^{-1} B \left( \frac{\alpha_0}{\Delta t} M + B^T A^{-1} B + D \right)^{-1} \left( F + \frac{1}{\Delta t} M \sum_{i=1}^m \alpha_i p_K^{n+1-i} \right) + C^T A^{-1} B \left( \frac{\alpha_0}{\Delta t}M + B^T A^{-1} B + D \right)^{-1} B^T A^{-1} F_v - C^T A^{-1} F_v \,, \end{array}$

Imposing that at each edge or face the hybrid unknown is single value we obtain a linear system for the hybrid unknown

$L \lambda^{n+1} = r \,.$

We recover the primal and dual variable as a post-process from the hybrid variable at element level, so we have

$\begin{array}{l} \displaystyle p_K^{n+1} = \left( \frac{\alpha_0}{\Delta t} M + B^T A^{-1} B + D\right)^{-1} \left( F + \frac{1}{\Delta t} M \sum_{i=1}^m \alpha_i p_K^{n+1-i} - B^T A^{-1} C \lambda_K^{n+1} \right)\,, \vspace{0.2cm} \\ \sigma_K^{n+1} = -A^{-1} \left( B p_K^{n+1} + C \lambda_K^{n+1} \right) \,. \end{array}$

Note: In the code we do not use the matrix and the vector , because all the boundary conditions are imposed via BCHandler class.; The initial time is not fixed at zero.; Example of usage can be found in darcy_nonlinear and darcy_linear. Coupled with an hyperbolic solver in impes.

Todo:: Insert any scientific publications that use this solver.

Definition at line 251 of file DarcySolverTransient.hpp.

Member Typedef Documentation

◆ mesh_Type

typedef MeshType mesh_Type

Typedef for mesh template.

Definition at line 261 of file DarcySolverTransient.hpp.

◆ darcySolverTransient_Type

typedef DarcySolverTransient< mesh_Type > darcySolverTransient_Type

Self typedef.

Definition at line 264 of file DarcySolverTransient.hpp.

◆ darcySolverLinear_Type

typedef DarcySolverLinear< mesh_Type > darcySolverLinear_Type

Darcy solver class.

Definition at line 267 of file DarcySolverTransient.hpp.

◆ data_Type

typedef darcySolverLinear_Type::data_Type data_Type

Typedef for the data type.

Definition at line 270 of file DarcySolverTransient.hpp.

◆ dataPtr_Type

typedef darcySolverLinear_Type::dataPtr_Type dataPtr_Type

Shared pointer for the data type.

Definition at line 273 of file DarcySolverTransient.hpp.

◆ scalarFctPtr_Type

typedef darcySolverLinear_Type::scalarFctPtr_Type scalarFctPtr_Type

Shared pointer to a scalar value function.

Definition at line 276 of file DarcySolverTransient.hpp.

◆ vectorPtr_Type

typedef darcySolverLinear_Type::vectorPtr_Type vectorPtr_Type

Shared pointer to a distributed vector.

Definition at line 279 of file DarcySolverTransient.hpp.

◆ vector_Type

typedef darcySolverLinear_Type::vector_Type vector_Type

Distributed vector.

Definition at line 282 of file DarcySolverTransient.hpp.

◆ scalarFieldPtr_Type

typedef darcySolverLinear_Type::scalarFieldPtr_Type scalarFieldPtr_Type

Shared pointer to a scalar field.

Definition at line 285 of file DarcySolverTransient.hpp.

◆ scalarField_Type

typedef darcySolverLinear_Type::scalarField_Type scalarField_Type

Scalar field.

Definition at line 288 of file DarcySolverTransient.hpp.

◆ matrixElementalContainer_Type

typedef std::vector< MatrixElemental > matrixElementalContainer_Type

Container of matrix elemental.

Definition at line 291 of file DarcySolverTransient.hpp.

◆ timeAdvance_Type

typedef TimeAdvanceBDF< vector_Type > timeAdvance_Type

Time advance scheme.

Definition at line 294 of file DarcySolverTransient.hpp.

◆ timeAdvancePtr_Type

typedef std::shared_ptr< timeAdvance_Type > timeAdvancePtr_Type

Shared pointer to a time advance scheme.

Definition at line 297 of file DarcySolverTransient.hpp.

Constructor & Destructor Documentation

◆ DarcySolverTransient() [1/2]

DarcySolverTransient ( )

Constructor for the class.

Definition at line 467 of file DarcySolverTransient.hpp.

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◆ ~DarcySolverTransient()

virtual ~DarcySolverTransient ( )

inlinevirtual

Virtual destructor.

Definition at line 308 of file DarcySolverTransient.hpp.

◆ DarcySolverTransient() [2/2]

DarcySolverTransient ( const darcySolverTransient_Type & )

private

Inhibited copy constructor.

Member Function Documentation

◆ setup()

void setup ( )

virtual

Set up the linear solver and the preconditioner for the linear system.

Reimplemented from DarcySolverLinear< MeshType >.

Reimplemented in DarcySolverTransientNonLinear< MeshType >.

Definition at line 488 of file DarcySolverTransient.hpp.

◆ solve()

void solve ( )

virtual

Solve the problem.

Reimplemented from DarcySolverLinear< MeshType >.

Reimplemented in DarcySolverTransientNonLinear< MeshType >.

Definition at line 552 of file DarcySolverTransient.hpp.

◆ setInitialPrimal()

void setInitialPrimal ( const scalarFctPtr_Type & primalInitialFct )

Initialize primal solution.

Set the initial value function for the primal variable and compute the primal variable at step zero.

Parameters

primalInitial The primal initial function.

Definition at line 526 of file DarcySolverTransient.hpp.

◆ setMass()

void setMass ( const scalarFctPtr_Type & massFct )

Set mass matrix.

Set the mass term, the default source term is the function one. By defaul it does not depend on time.

Parameters

mass	Mass term for the problem.

Loop on all the volume elements.

Definition at line 573 of file DarcySolverTransient.hpp.

◆ localMatrixComputation()

void localMatrixComputation	(	const UInt &	iElem,
		MatrixElemental &	elmatMix,
		MatrixElemental &	elmatReactionTerm
	)

protectedvirtual

Compute element matrices.

Call the Darcy solver localMatrixComputation method and compute the mass matrix for the time dependent term.

Parameters

iElem	Id of the current geometrical element.
elmatMix	The local matrix in mixed form.
elmatReactionTerm	The local matrix for the reaction term.

Reimplemented from DarcySolverLinear< MeshType >.

Reimplemented in DarcySolverTransientNonLinear< MeshType >.

Definition at line 624 of file DarcySolverTransient.hpp.

◆ localVectorComputation()

void localVectorComputation	(	const UInt &	iElem,
		VectorElemental &	elvecMix
	)

protectedvirtual

Computes local vectors.

Call the Darc_y solver localVectorComputation method and compute the additional scalar vector for the time dependent term.

Parameters

iElem	Id of the current geometrical element.
elvecMix	The local vector in mixed form.

Reimplemented from DarcySolverLinear< MeshType >.

Reimplemented in DarcySolverTransientNonLinear< MeshType >.

Definition at line 650 of file DarcySolverTransient.hpp.

◆ postComputePrimalAndDual()

virtual void postComputePrimalAndDual ( )

inlineprotectedvirtual

Do some computation after the calculation of the primal and dual variable.

Save into the time advance scheme the computed primal solution.

Reimplemented from DarcySolverLinear< MeshType >.

Definition at line 378 of file DarcySolverTransient.hpp.

◆ setupTime()

void setupTime ( )

protected

Setup the time data.

Definition at line 503 of file DarcySolverTransient.hpp.

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◆ operator=()

darcySolverTransient_Type& operator= ( const darcySolverTransient_Type & )

private

Inhibited assign operator.

Field Documentation

◆ M_timeAdvance

timeAdvancePtr_Type M_timeAdvance

protected

Time advance.

Definition at line 394 of file DarcySolverTransient.hpp.

◆ M_rhsTimeAdvance

vectorPtr_Type M_rhsTimeAdvance

protected

Right hand side coming from the time advance scheme.

Definition at line 397 of file DarcySolverTransient.hpp.

◆ M_primalFieldInitialFct

scalarFctPtr_Type M_primalFieldInitialFct

private

Initial time primal variable.

Definition at line 426 of file DarcySolverTransient.hpp.

◆ M_massFct

scalarFctPtr_Type M_massFct

private

Mass function, it does not depend on time.

Definition at line 429 of file DarcySolverTransient.hpp.

◆ M_localMassMatrix

matrixElementalContainer_Type M_localMassMatrix

private

Local mass matrices.

Definition at line 438 of file DarcySolverTransient.hpp.

◆ M_reusePrec

bool M_reusePrec

private

Boolean that indicates if the preconditioner is re-used or not.

Definition at line 447 of file DarcySolverTransient.hpp.

◆ M_updated

bool M_updated

private

Boolean that indicates if the matrix is updated for the current iteration.

Definition at line 450 of file DarcySolverTransient.hpp.

◆ M_maxIterSolver

UInt M_maxIterSolver

private

Interger storing the max number of solver iteration with preconditioner recomputing.

Definition at line 453 of file DarcySolverTransient.hpp.

◆ M_recomputeMatrix

bool M_recomputeMatrix

private

Boolean that indicates if the matrix is recomputed for the current iteration.

Definition at line 456 of file DarcySolverTransient.hpp.

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

DarcySolverTransient.hpp

Public Types

Constructors & destructor

methods

Update and set methods

Protected methods

Time advance stuff

Private Constructors

Private Operators

Data of the problem

Algebraic stuff

Additional Inherited Members

Detailed Description

template<typename MeshType> class LifeV::DarcySolverTransient< MeshType >

Member Typedef Documentation

◆ mesh_Type

◆ darcySolverTransient_Type

◆ darcySolverLinear_Type

◆ data_Type

◆ dataPtr_Type

◆ scalarFctPtr_Type

◆ vectorPtr_Type

◆ vector_Type

◆ scalarFieldPtr_Type

◆ scalarField_Type

◆ matrixElementalContainer_Type

◆ timeAdvance_Type

◆ timeAdvancePtr_Type

Constructor & Destructor Documentation

◆ DarcySolverTransient() [1/2]

◆ ~DarcySolverTransient()

◆ DarcySolverTransient() [2/2]

Member Function Documentation

◆ setup()

◆ solve()

◆ setInitialPrimal()

◆ setMass()

◆ localMatrixComputation()

◆ localVectorComputation()

◆ postComputePrimalAndDual()

◆ setupTime()

◆ operator=()

Field Documentation

◆ M_timeAdvance

◆ M_rhsTimeAdvance

◆ M_primalFieldInitialFct

◆ M_massFct

◆ M_localMassMatrix

◆ M_reusePrec

◆ M_updated

◆ M_maxIterSolver

◆ M_recomputeMatrix

template<typename MeshType>
class LifeV::DarcySolverTransient< MeshType >