diffusion Module Reference

MODULE FOR DIFFUSION PROBLEMS More...

Data Types
interface	l2_product
	L2 product of a function with the basis functions. More...

Functions/Subroutines
subroutine, public	diffusion_matrix_pattern (g, X_h, qdm)
	Define the sparsity pattern for diffusion matrices More...
subroutine, public	diffusion_massmat (mass, a, X_h, qdm, dofToDof)
	Assemble the mass matrix of the bilinear product: More...
subroutine, public	diffusion_stiffmat (stiff, b, X_h, qdm, dofToDof)
	Assemble the stiffness matrix of the bilinear product: More...
subroutine, public	diffusion_neumann_rhs (rhs, g, X_h, quad_type, f)
	L2 scalar product of a scalar function \( g \) with the basis functions fo the finite element space \( X_h \) on \(\Gamma_f \subset \partial\Omega\) a part of the domain boundary. More...
subroutine, public	diffusion_dirichlet (K, rhs, g, X_h, rho, f)
	DIRICHLET BOUNDARY CONDITION FOR A DIFFUSION PROBLEM More...
subroutine	fespace_l2_product (FV, f, X_h, qdm)
	L2 Product of a scalar function \( f:~\R^3 \mapsto \R\) with the basis functions of a scalar finite element space. More...
subroutine, public	diffusion_massmat_vect (mass, b, X_h, qdm, dofToDof)
	Assemble the mass matrix of the bilinear product: More...
subroutine, public	diffusion_mixed_divmat (divMat, X_s, X_v, qdm)
	Assemble the matrix of the bilinear product: More...

Detailed Description

MODULE FOR DIFFUSION PROBLEMS

A module for diffusion equations of the form

\(~~~~~~~~~~~~~~~~~~\) \(~~~~~~~~~ -\dv(B(x) \nabla u) + \rho(x) u = f \)

with \( B(x)\) a diffusivity tensor.

Tutorial examples:

• poisson_2d.f90: Poisson problem in 2D.
• diffusion_Neumann.f90: a general diffusion equation with Neumann boundary condition.
• diffusion_Dirichlet_Neumann.f90: a general diffusion equation with mixed Dirichlet-Neumann boundary condition.

Author

Charles Pierre

Function/Subroutine Documentation

diffusion_dirichlet()

subroutine, public diffusion::diffusion_dirichlet	(	type(csr), intent(inout)	K,
		real(rp), dimension(:), intent(inout)	rhs,
		procedure(r3tor)	g,
		type(fespace), intent(in)	X_h,
		real(rp), intent(in)	rho,
		procedure(r3tor), optional	f
	)

DIRICHLET BOUNDARY CONDITION FOR A DIFFUSION PROBLEM

Given:

• the boundary sub-domain

  \[ \Gamma_f = \partial\Omega \cap \{f\ge 0\}, \]

• the source term \( g \) on \( \Gamma_f \).
• the linear ssytem

  \[ K U = \text{rhs}, \]

• the penalty coefficient \( \rho \gg 1 \),

Penalise the linear system on the lines i associated with a finite element node \( x\in \Gamma_f \):

• rhs(i) = rhs(i) + \( \rho g(x)\)
• \( K_{i,i} = K_{i,i} + \rho \)

INPUT:

• \( K\), rhs the original matrix and right hand side of the system,
• \( g:~\R^3 \mapsto \R \)

• \( X_h \) a scalar finite element space on a mesh of the domain \( \Omega \)

• rho = penalty coefficient$

• \( f:~\R^3 \mapsto \R \) a function to characterise the boundary sub-domain \(\Gamma_f = \partial\Omega \cap \{f\ge 0\} \).
  This is an optional argument, if not provided \( \Gamma_f = \partial\Omega \).

OUTPUT:

• \( K\), rhs the penalised matrix and right hand side of the system,

Definition at line 656 of file diffusion.F90.

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diffusion_massmat()

subroutine, public diffusion::diffusion_massmat	(	type(csr), intent(inout)	mass,
		procedure(r3tor)	a,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm,
		type(graph), intent(in), optional	dofToDof
	)

Assemble the mass matrix of the bilinear product:

\( (u,v) \mapsto \int_O a(x)~ u(x) v(x) \dx ~~\in\R\)

• \( u, v \in X_h = \) scalar finite element space on the mesh \( \T \).

• \( O \) = integration domain, see quadmesh_mod for a definition.

OUTPUT:

• mass= mass matrix

INPUT :

• \( a:~x \in\R^3 \mapsto a(x)\in\R \), density function

• \( X_h =\) scalar finite element space on the mesh \( \T \)

• qdm = integration method on the mesh \( \T \)

• dofToDof = connectivity graph DOF –> DOF (optional, of type graph_mod::graph)

Definition at line 168 of file diffusion.F90.

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diffusion_massmat_vect()

subroutine, public diffusion::diffusion_massmat_vect	(	type(csr), intent(inout)	mass,
		procedure(r3xr3xr3tor)	b,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm,
		type(graph), intent(in), optional	dofToDof
	)

Assemble the mass matrix of the bilinear product:

\( (p,q) \mapsto \int_O b(x, p(x), q(x)) \dx ~~\in\R\)

• \( p, q \in X_h = \) vector finite element space on the mesh \( \T \).

• \( O \) = integration domain, see quadmesh_mod for a definition.

OUTPUT:

• mass= mass matrix

INPUT :

• \( b:~ (x, \xi_1, \xi_2) \in \R^3 \times \R^3 \times \R^3 \mapsto b(x, \xi_1, \xi_2) \in \R \)
  assumed to be bilinear and symmetric in \(\xi_1\) and \( \xi_2\).
  This assumption must be respected, it is not checked.

• \( X_h =\) vector finite element space on the mesh \( \T \) (of type fespacexk_mod::fespacexk)

• qdm = integration method on the mesh \( \T \) (of type quadmesh_mod::quadmesh)

• dofToDof = connectivity graph DOF –> DOF (optional, of type graph_mod::graph)

Definition at line 899 of file diffusion.F90.

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diffusion_matrix_pattern()

subroutine, public diffusion::diffusion_matrix_pattern	(	type(graph), intent(inout)	g,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm
	)

Define the sparsity pattern for diffusion matrices

It is given by the connectivity graph Dof –> Dof:
between the degrees of freedom of the finite element space \( X_h \) on the mesh \( \T \).

DOF_i is in relation with DOF_j if:

• they are asociated to a common cell of the mesh \( \T \),

• and if that common cell is associated to a non-void quadrature rule in the integration method.

OUTPUT:

• g = connevtivity graph

INPUT :

• \( X_h \) finite element space on the mesh \( \T \)

• qdm = integration method on the mesh \( \T \)

Definition at line 96 of file diffusion.F90.

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diffusion_mixed_divmat()

subroutine, public diffusion::diffusion_mixed_divmat	(	type(csr), intent(inout)	divMat,
		type(fespace), intent(in)	X_s,
		type(fespace), intent(in)	X_v,
		type(quadmesh), intent(in)	qdm
	)

Assemble the matrix of the bilinear product:

\( (u,p) \mapsto \int_O u(x) \dv p(x) \dx ~~\in\R\)

• \( u \in X_s = \) scalar finite element space on the mesh \( \T \).

• \( p \in X_v = \) vector finite element space on the mesh \( \T \).

• \( O \) = integration domain, see quadmesh_mod for a definition.

OUTPUT:

• divMat= matrix of the product

INPUT :

• \( X_s =\) scalar finite element space on the mesh \( \T \)

• \( X_v =\) vector finite element space on the mesh \( \T \)

• qdm = integration method on the mesh \( \T \)

Definition at line 1164 of file diffusion.F90.

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diffusion_neumann_rhs()

subroutine, public diffusion::diffusion_neumann_rhs	(	real(rp), dimension(:), allocatable	rhs,
		procedure(r3tor)	g,
		type(fespace), intent(in)	X_h,
		integer, intent(in)	quad_type,
		procedure(r3tor), optional	f
	)

L2 scalar product of a scalar function \( g \) with the basis functions fo the finite element space \( X_h \) on \(\Gamma_f \subset \partial\Omega\) a part of the domain boundary.

INPUT:

• \( g:~\R^3 \mapsto \R \)

• \( X_h \) a scalar finite element space on a mesh of the domain \( \Omega \)

• 'quad_type' quadrature rule for the boundary cells \( K \subset \Gamma_f\)

• \( f:~\R^3 \mapsto \R \) a function to characterise the boundary sub-domain \(\Gamma_f = \partial\Omega \cap \{f\ge 0\} \).
  This is an optional argument, if not provided \( \Gamma_f = \partial\Omega \).

OUTPUT:

• rhs = \( (g_i)_{1\le i\le N} \) with \( g_i = \int_{\Gamma_f} g(x) v_i \dx \), where \( (v_i)_{1\le i\le N} \) are the basis functions of \( X_h\).

Definition at line 590 of file diffusion.F90.

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diffusion_stiffmat()

subroutine, public diffusion::diffusion_stiffmat	(	type(csr), intent(inout)	stiff,
		procedure(r3xr3xr3tor)	b,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm,
		type(graph), intent(in), optional	dofToDof
	)

Assemble the stiffness matrix of the bilinear product:

\( (u,v) \mapsto \int_O b(x, \nabla u(x), \nabla v(x)) \dx ~~\in\R\)

• \( u, v \in X_h = \) scalar finite element space on the mesh \( \T \).

• \( O \) = integration domain, see quadmesh_mod for a definition.

OUTPUT:

• stiff= stiffness matrix

INPUT :

• \( b:~ (x, \xi_1, \xi_2) \in \R^3 \times \R^3 \times \R^3 \mapsto b(x, \xi_1, \xi_2) \in \R \)
  assumed to be bilinear and symmetric in \(\xi_1\) and \( \xi_2\).
  This assumption must be respected, it is not checked.

• \( X_h =\) scalar finite element space on the mesh \( \T \) (of type fespacexk_mod::fespacexk)

• qdm = integration method on the mesh \( \T \) (of type quadmesh_mod::quadmesh)

• dofToDof = connectivity graph DOF –> DOF (optional, of type graph_mod::graph)

NOTE : every cell \(K \subset O\) is either

• an open domain in \(\R^3\)
• a parametrised surface in \(\R^3\)
• a parametrised curve in \(\R^3\)

In the 2nd and 3rd cases : \( \nabla u(x)\) is in the tangent gradient (in the tangent space).

Definition at line 366 of file diffusion.F90.

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fespace_l2_product()

subroutine diffusion::fespace_l2_product ( real(rp), dimension(:), allocatable  FV, procedure(r3tor)  f, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

L2 Product of a scalar function \( f:~\R^3 \mapsto \R\) with the basis functions of a scalar finite element space.

INPUT

• \( f:~ \R^3 \mapsto \R\)

• \( X_h =\) scalar finite element space on the mesh \( \T \)

• qdm = integration method on the mesh \( \T \)

OUTPUT:

• \( FV = (FV_i)_{1\le i\le N} \) with \( FV_i = \int_O f(x) v_i(x) \dx = [ (f,u_i)_{{\rm L}^2(O)} ] \)
  where the \( (v_i)_{1\le i\le N} \) are the basis function of \( X_h \).

• \( O \) = integration domain, see quadmesh_mod for a definition.

Definition at line 741 of file diffusion.F90.

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