integral Module Reference

COMPUTATION OF INTEGRALS using a quadrature methods on a mesh. More...

Data Types
interface	integ
	Integration of scalar functions. More...
interface	l2_dist
	integral L2 distance More...
interface	l2_dist_grad

Functions/Subroutines
real(rp) function	integ_scal_func (u, qdm)
	Returns \( \int_O u(x) \dx \). More...
real(rp) function	integ_scal_fe (E, u, uh, X_h, qdm)
	Returns \( \int_O E(x, u(x), u_h(x)) \dx \). More...
real(rp) function	integ_scal_fe_grad (E, phi, uh, X_h, qdm)
	Returns \( \int_O E(x, \phi(x), \nabla u_h(x)) \dx \). More...
real(rp) function, public	integ_scal_fe_grad_proj (E, phi, uh, X_h, qdm)
	Returns \( \int_O E(x, \phi_T(x), \nabla u_h(x)) \dx \). More...
real(rp) function	integ_scal_fe_grad_2 (E, uh1, uh2, X_h, qdm)
	Returns \( \int_O E(x, \nabla u1_h(x), \nabla u2_h(x)) \dx \). More...
real(rp) function	e_l2 (x, u1, u2)
	\( (x-y)^2 \) More...
real(rp) function	e_l2_vect (x, p1, p2)
	\( (p-q)^2 \) More...
real(rp) function	l2_dist_scalfe (u, uh, X_h, qdm)
	Returns \( \int_O \vert u - u_h \vert^2 \dx \). More...
real(rp) function	l2_dist_grad_scalfe (phi, uh, X_h, qdm)
	Returns \( \int_O \vert \phi - \nabla u_h \vert^2 \dx \). More...
real(rp) function, public	l2_dist_grad_proj (phi, uh, X_h, qdm)
	Returns \( \int_O \vert \phi_T - \nabla u_h \vert^2 \dx \). More...
real(rp) function, public	l2_dist_vect (phi, phi_h, X_h, qdm)
	Returns \( \int_O \vert \phi - \phi_h \vert^2 \dx \). More...

Detailed Description

COMPUTATION OF INTEGRALS using a quadrature methods on a mesh.

Given:

• a mesh \( \mathcal{T} \) of a domain \( \Omega \),
• an integration method quadMesh on \( \mathcal{T}\),

This module proposes various computation of integrals with an integration domain composed of cells of the mesh \( \mathcal{T} \).

Integration domain: see quadmesh_mod mreamble for a definition.

Author

Charles Pierre

Function/Subroutine Documentation

e_l2()

real(rp) function integral::e_l2 ( real(rp), dimension(3), intent(in)  x, real(rp), intent(in)  u1, real(rp), intent(in)  u2  )

private

\( (x-y)^2 \)

Definition at line 875 of file integral.F90.

e_l2_vect()

real(rp) function integral::e_l2_vect ( real(rp), dimension(3), intent(in)  x, real(rp), dimension(3), intent(in)  p1, real(rp), dimension(3), intent(in)  p2  )

private

\( (p-q)^2 \)

Definition at line 884 of file integral.F90.

integ_scal_fe()

real(rp) function integral::integ_scal_fe ( procedure(r3xrxrtor)  E, procedure(r3tor)  u, real(rp), dimension(:), intent(in)  uh, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O E(x, u(x), u_h(x)) \dx \).

• \( E~: \R^3 \times \R \times \R \mapsto \R \)

• \(u~: \R^3 \mapsto \R \) is a scalar function,

• \( u_h~: \Omega \mapsto \R\) is a scalar finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 193 of file integral.F90.

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integ_scal_fe_grad()

real(rp) function integral::integ_scal_fe_grad ( procedure(r3xr3xr3tor)  E, procedure(r3tor3)  phi, real(rp), dimension(:), intent(in)  uh, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O E(x, \phi(x), \nabla u_h(x)) \dx \).

• \( E~: \R^3 \times \R^3 \times \R^3 \mapsto \R \)

• \(\phi~: \R^3 \mapsto \R^3 \) is a vector function,

• \( u_h~: \Omega \mapsto \R\) is a scalar finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 345 of file integral.F90.

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integ_scal_fe_grad_2()

real(rp) function integral::integ_scal_fe_grad_2 ( procedure(r3xr3xr3tor)  E, real(rp), dimension(:), intent(in)  uh1, real(rp), dimension(:), intent(in)  uh2, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O E(x, \nabla u1_h(x), \nabla u2_h(x)) \dx \).

• \( E~: \R^3 \times \R^3 \times \R^3 \mapsto \R \)

• \( ui_h~: \Omega \mapsto \R\) is a scalar finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 702 of file integral.F90.

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integ_scal_fe_grad_proj()

real(rp) function, public integral::integ_scal_fe_grad_proj	(	procedure(r3xr3xr3tor)	E,
		procedure(r3tor3)	phi,
		real(rp), dimension(:), intent(in)	uh,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm
	)

Returns \( \int_O E(x, \phi_T(x), \nabla u_h(x)) \dx \).

same as integral::integ_scal_fe_grad BUT :

\( \phi_T \): tangent component of \( \phi\) in the tangent space to any cell in the integration domain O

Definition at line 511 of file integral.F90.

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integ_scal_func()

real(rp) function integral::integ_scal_func ( procedure(r3tor)  u, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O u(x) \dx \).

• \( u \) = scalar function,
• 'qdm' = integration method on a mesh,
  \( O \) is the integration domain, see quadmesh_mod for a definition.

Definition at line 81 of file integral.F90.

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l2_dist_grad_proj()

real(rp) function, public integral::l2_dist_grad_proj	(	procedure(r3tor3)	phi,
		real(rp), dimension(:), intent(in)	uh,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm
	)

Returns \( \int_O \vert \phi_T - \nabla u_h \vert^2 \dx \).

same as integral::l2_dist_grad BUT :

\( \phi_T \): tangent component of \( \phi\) in the tangent space to any cell in the integration domain O

Definition at line 956 of file integral.F90.

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l2_dist_grad_scalfe()

real(rp) function integral::l2_dist_grad_scalfe ( procedure(r3tor3)  phi, real(rp), dimension(:), intent(in)  uh, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O \vert \phi - \nabla u_h \vert^2 \dx \).

• \(\phi~: \R^3 \mapsto \R^3 \) is a vector function,

• \( u_h~: \Omega \mapsto \R\) is a scalar finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 934 of file integral.F90.

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l2_dist_scalfe()

real(rp) function integral::l2_dist_scalfe ( procedure(r3tor)  u, real(rp), dimension(:), intent(in)  uh, type(fespace), intent(in)  X_h, type(quadmesh), intent(in)  qdm  )

private

Returns \( \int_O \vert u - u_h \vert^2 \dx \).

• \( u~: \R^3 \mapsto \R \) is a scalar function,

• \( u_h~: \Omega \mapsto \R\) is a scalar finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 905 of file integral.F90.

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l2_dist_vect()

real(rp) function, public integral::l2_dist_vect	(	procedure(r3tor3)	phi,
		real(rp), dimension(:), intent(in)	phi_h,
		type(fespace), intent(in)	X_h,
		type(quadmesh), intent(in)	qdm
	)

Returns \( \int_O \vert \phi - \phi_h \vert^2 \dx \).

• \( \phi~: \R^3 \mapsto \R^3 \) is a vector function,

• \( \phi_h~: \Omega \mapsto \R^3\) is a vector finite element function in the finite element space \( X_h \)
  (associated to a mesh \( \T \) with domain \( \Omega \)),

• 'qdm' = integration method on the mesh \( \T \),

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

Definition at line 986 of file integral.F90.

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