DarcyIntegrators Namespace Reference

Namespaces
	Polynomial

Classes
class	ErrorIntegrator
class	Estimator
class	IdentityTensorFunction
class	Interpolator
class	Matrix
class	NoForceResidual
class	Postprocessor
class	RHSIntegrator
class	SystemIntegrator

Functions
template<int dim>
void	weighted_mass_matrix (dealii::FullMatrix< double > &M, const dealii::FEValuesBase< dim > &fe, const dealii::TensorFunction< 2, dim > &weight)

Detailed Description

Namespace containing integrators for mixed discretizations of Darcy's equation.

Integrators for Darcy equation

Namespace containing local integrator classes to integrate the right hand side, residual and error for a Darcy problem, where the exact solution is a known polynomial.

The equations solved are the Darcy equations (with all coefficients equal to one)

\begin{align*} u - \nabla p &= f \\ \nabla\cdot u &= g \end{align*}

The solutions in this namespace are parameterized by three tensor product polynomials \(\psi\), \(\phi\) and \(\pi\) which all have the form

\[ p(x,y) = p_{1d}(x)*p_{1d}(y). \]

Given these three polynomials, the solutions are given by the equations

\begin{align*} u &= \nabla \times \psi + \nabla \phi \\ p &= \phi - \pi \end{align*}

The corresponding right hand side is given by

\begin{align*} f &= \nabla \times \psi + \nabla \phi \\ g &= \Delta \phi \end{align*}

Function Documentation

template<int dim>

void DarcyIntegrators::weighted_mass_matrix	(	dealii::FullMatrix< double > &	M,
		const dealii::FEValuesBase< dim > &	fe,
		const dealii::TensorFunction< 2, dim > &	weight
	)

The mass matrix for a weighted L2 inner product of vector valued finite elements, i.e.

\[ M_{ij} = \int K \phi_j \phi_i \]

where \(K\) is a tensor valued weight and \((\phi_i)\) are the local shape functions.