SFEMaNS  version 4.1 (work in progress) Reference documentation for SFEMaNS
doc_intro.h File Reference

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Functions

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic f with f (r,\theta, z)\f$the cylindrical coordinates

Variables

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode
independently

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic form section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic
theta

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic z section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic
t

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f$f_h

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f sin_m section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f
Omega_ {T}^{2D}\f$section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f f
f
Omega_v

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f left section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as
follows

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f$n geq0 f let f
bu {n+1}\f$section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that matches the Dirichlet boundary conditions of the problem section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f textbf section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in
the following generic f with f
$t f the time and f$M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text Function Documentation  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f ( r , \ theta, z )

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Variable Documentation

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f$f_h
Initial value:
{m,\cos}\f$and \f$f_h^{m,\sin}\f$can be approximated independtly modulo the computation of nonlinear terms. Introducing the functions \f$\cos_m = \cos(m\theta)\f$section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic theta
Definition: doc_intro.h:193
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f$the cylindrical coordinates
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f$f_h
Definition: doc_intro.h:204

Definition at line 204 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments • For f$n geq0 f let f bu {n+1}\f$• Definition at line 327 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as follows

Definition at line 218 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic form Definition at line 193 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode independently Definition at line 193 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f left

Definition at line 218 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f Omega_ {T}^{2D}\f$

Definition at line 214 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f Omega_v Initial value: {2D}\f$ and \f$\Omega^{2D}\f$ of
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f Omega_ Definition: doc_intro.h:214 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates Definition at line 215 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments
• For f $n geq0 f let f that matches the Dirichlet boundary conditions of the problem • Definition at line 327 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f sin_m
Initial value:
= \sin(m\theta)\f$and a basis functions \f$(\phi_j)_{j \in J}\f$of the finite element space of the meridian section results in \f$(\phi_j \cos_m)_{j\in J, m \in [|0,M|]} \cup (\phi_j \sin_m)_{j\in J, m \in [|1,M|]}\f$being a basis of the space of approximation. @subsection doc_intro_fram_work_num_app_space Space of approximation We set the number of Fourier mode to \f$M\f$. We define the meridian sections \f$\Omega_{c,f}^{2D}\f$section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f Omega_
Definition: doc_intro.h:214
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta Definition: doc_intro.h:193 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f sin_m
Definition: doc_intro.h:207

Definition at line 207 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic t Initial value: =f_h^{0,\cos}(r,z,t) + \sum_{m=1}^M f_h^{m,\cos} \cos(m\theta) + f_h^{m,\sin} \sin(m\theta) section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic t
Definition: doc_intro.h:199
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta Definition: doc_intro.h:193 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic z
Definition: doc_intro.h:193

Definition at line 199 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments • For f$n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text
• Initial value:
{c}_\text{div}\f$is a penalty coefficent section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text Definition: doc_intro.h:342 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates Definition at line 342 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f f textbf Initial value: {v}\f$ in \f$\textbf{V}_{h,0}^\bu\f$:
\f{equation}{
\label{eq:SFEMaNS_weak_from_NS_1}
\int_{\Omega_{c,f}} \frac{3}{2 \tau} \textbf{u}^{n+1} \cdot \textbf{v}
+ \text{c}_\text{div} h^{-1} \DIV\bu^{n+1} \DIV \bv=
- \int_{\Omega_{c,f}} ( \frac{-4 \textbf{u}^n
+ \textbf{u}^{n-1}}{2 \tau}
+ \GRAD ( p^n +\frac{4\psi^n - \psi^{n-1}}{3} ) ) \cdot \textbf{v} \
+ \int_{\Omega_{c,f}} ( \textbf{f}^{n+1} - (\ROT \textbf{u}^{*,n+1} )
\times \textbf{u}^{*,n+1} ) \cdot \textbf{v} ,
where \f$h\f$ is the local mesh size
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f$t f the time and f $M f the number of Fourier modes considered The unknown f f f Omega_ Definition: doc_intro.h:214 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text Definition: doc_intro.h:342 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f bu Definition: doc_intro.h:327 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic f with f $t f the time and f$M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f textbf Definition: doc_intro.h:330 Definition at line 330 of file doc_intro.h.  section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic theta

Definition at line 193 of file doc_intro.h.

 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f \$f f is written in the following generic z

Definition at line 193 of file doc_intro.h.