step7
Compare DPG Error Indicators and Actual Error

step7, a FENICS code which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem, for which an exact solution is known, in order to compare the DPG error indicators with the exact error.

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/step7
2. fenicsproject run
3. python3 step7.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:

The program can run 3 different test cases, and a variety of mesh sizes. Data is listed here only for the third test case, and a (2x)16x16 mesh.

• step7.py applies the DPG method to a Poisson problem.
• step7_case3.txt the output file for meshes 4, 8 and 16.
• step7_indicators_16.png a plot of the DPG error indicators.
• step7_mesh_16.png a plot of the mesh.
• step7_u_16.png a plot of the solution.
• step7_u_exact_16.png a plot of the exact solution.