step3
Discontinuous Petrov Galerkin for Poisson Problem on Unit Square

step3, a FENICS code which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem over the unit square with piecewise constant diffusivity kappa(x,y). In addition, it computes and plots the value of |grad(u)|, the norm of the gradient of the solution.

The program relies heavily on the ideas and implementation embodied in Jay Gopalakrishnan's dpg_laplace code.

Note that I have installed FENICS using Docker, and so to run this script I issue the commands:

1. cd $HOME/fenicsproject/step3
2. fenicsproject run
3. python3 step3.py
4. exit

Licensing:

The GNU LGPL license.

Related Data and Programs:

dpg_bvp, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a boundary value problem over an interval, by Jay Gopalakrishnan.

dpg_laplace, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem over the unit square, by Jay Gopalakrishnan.

dpg_laplace_adapt, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem over the unit square, with adaptivity, by Jay Gopalakrishnan.

Reference:

1. Jay Gopalakrishnan,
   Five lectures on DPG Methods,
   Spring 2013, Portland State University,
   arXiv:1306.0557v2 [math.NA] 28 Aug 2014.

Source Code:

• step3.py applies the DPG method to a Poisson problem.
• step3.txt the output file.
• step3_indicators.png a plot of the DPG error indicators.
• step3_kappa.png a plot of the diffusivity kappa(x,y).
• step3_mesh.png a plot of the mesh.
• step3_u.png a plot of the solution.
• step3_u_grad_norm.png a plot of the norm of the gradient of the solution.