stokes_2d_exact

stokes_2d_exact, a Python code which evaluates exact solutions (u,v,p)(x,y) to the incompressible steady Stokes equations in 2D, which are a system of partial differential equations (PDE) of the form: -uxx-uyy+px=f, -vxx-vyy+py=g, ux+vy=h.

The incompressible steady 2D Stokes equations seek functions u(x,y), v(x,y), (the velocity vector) and p(x,y) (the pressure) such that

        - uxx - uyy + dpdx = f
        - vxx - vyy + dpdy = g
          ux  + vy         = h

while satisfying boundary conditions for u and v, and a normalization condition for p, such as specifying its value at one point, or requiring that the integral of p over the region be 0.

Three exact solution fields are provided here.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

stokes_2d_exact is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

python_exact, a Python code which evaluates exact solutions to a few selected examples of ordinary differential equations (ODE) and partial differential equations (PDE).

Reference:

Junping Wang, Yanqiu Wang, Xiu Ye,
A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods,
SIAM Journal on Scientific Computing,
Volume 31, Number 4, 2009, pages 2784-2802.
Howard Elman, Alison Ramage, David Silvester,
Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics,
Oxford, 2005,
ISBN: 978-0198528678,
LC: QA911.E39.

Source Code:

stokes_2d_exact.py, the source code.
stokes_2d_exact.sh, runs all the tests.
stokes_2d_exact.txt, the output file.

Graphics files for flow #1:

stokes1_commands.txt, commands to draw the velocity field.
stokes1_data.txt, velocity field data.
stokes1.png, a PNG plot of the velocity field.
stokes1_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Graphics files for flow #2:

stokes2_commands.txt, commands to draw the velocity field.
stokes2_data.txt, velocity field data.
stokes2.png, a PNG plot of the velocity field.
stokes2_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Graphics files for flow #3:

stokes3_commands.txt, commands to draw the velocity field.
stokes3_data.txt, velocity field data.
stokes3.png, a PNG plot of the velocity field.
stokes3_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Last revised on 12 February 2015.