A self-contained 2D finite element examples using Python and scipy and matplotlib.

The codes py_distmesh2d.py and mesh_examples.py are from py_distmesh2d by Constantine Khroulev. They are a Python re-implementation of distmesh2d in P.-O. Persson and G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004 (http://persson.berkeley.edu/distmesh/).

What's new?:

• meshtools.py: Python re-implementations of codes from the Math 692 one-page Matlab FEM code challenge, which dates to Fall 2004. These tools include fixmesh, edgelist, and bdyrefine

• cg.py: Boring implementation of the conjugate gradient method. (See H. Elman, D. Silvester, and A. Wathen, Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics. Oxford 2005.) This way we don't even need numpy.linalg.solve. In practice, performance for PDE problems here (as opposed to variational inequality problems) is limited by meshing, not linear algebra.

• poisson.py: Assemble and solve Poisson equation del^2 u = f. See also http://www.cs.uaf.edu/~bueler/poissonv2.pdf

• obstacle.py: Solve the obstacle problem associated to the Poisson equation. That is, solve the variational inequality. See also http://www.dms.uaf.edu/~bueler/obstacleDOC.pdf

Demos:

$ python fem_examples.py

gives the first solution plot, and

$ python obstacle_example.py

gives the second and third plots. The second plot shows the solution with a spherical obstacle. The third plot shows in red those nodes which meet (touch) the obstacle.