# Finite Elements for Fluids

## Instructor : Prof. Sheu

## Finite Element Method with FreeFem++

FreeFem++FreeFem++-cs

Documentation

March 10, 2016

March 31, 2016

April 7, 2016

May 5, 2016

## Triangular finite elements

March 10, 2016

Using the files

`triangles.dat`

and `dof.dat`

produced by the FreeFem++ script `basis.edp`

, we use GNUplot to exhibit the deegree of freedoms with this command `splot 'triangles.dat' w l notitle, 'dof.dat' t 'Deegrees of freedom' pt 6 lw 3`## Finite Element Mehod in FreeFem++

March 31, 2016

mesh.edp

elliptic.edp - Dirichlet type B.C.

elliptic.edp - Robin type B.C.

order.edp - for structured meshes

order.edp - for unstructured meshes

heat.edp - Dirichlet type B.C.

heat.edp - Dirichlet/Neumann mixed type B.C.

Homework correction

elliptic.edp - Dirichlet type B.C.

elliptic.edp - Robin type B.C.

order.edp - for structured meshes

order.edp - for unstructured meshes

heat.edp - Dirichlet type B.C.

heat.edp - Dirichlet/Neumann mixed type B.C.

Homework correction

Order of convergence between the computed finite element solution \( u_h \) and the exact solution \( u \).

Solution \( u_h \) of the elliptic problem with Robin oundary conditions in \( \Omega=[0,1]×[0,1] \) :

\(- \nabla^2 u +c\,u= f \text{ in } \Omega \)

\( \frac{\partial u}{\partial n} +\varepsilon u = 0 \text{ on } \partial\Omega \)

## Convection-diffusion / Stokes equations

April 7, 2016

Convection dominant solution of the convection-diffusion equation in the unit square.

Solution of the Stokes equations in a vessel with a stenose

## Characteristics-Galerkin method / Navier-Stokes equations

May 5, 2016