step6
Direct Solution of p-Laplacian Equation

step6, a FENICS code which solves the nonlinear p-Laplacian problem directly.

The p-Laplacian equation has the form

       - div kappa(u) grad u = f in Omega = [0,1]x[0,1]
                           u = 0 on dOmega

       kappa(u) = ( epsilon^2 + |grad u|^2) ) ^ ( (p-2)/2 )
                = ( epsilon^2 + wx^2 + wy^2 ) ^ ( (p-2)/2 )

       epsilon = 1.0E-05

       f(x,y) = 2*pi*pi*sin(pi*x) * sin(pi*y)

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/step6
2. fenicsproject run
3. python3 step6.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. John Barrett, W B Liu,
   Finite Element Approximation of the p-Laplacian,
   Mathematics of Computation,
   Volume 61, Number 204, pages 423-537, October 1993.
2. Patrick Farrell, Hans Petter Langtangen, Marie Rognes, Garth Wells,
   Implementing finite element models in Python/FEniCS: steady nonlinear PDEs,
   MMSC: Python in Scientific Computing,
   May 6, 2015
3. Jay Gopalakrishnan,
   Five lectures on DPG Methods,
   Spring 2013, Portland State University,
   arXiv:1306.0557v2 [math.NA] 28 Aug 2014.

Source Code:

• step6.py applies the DPG method to a Poisson problem.
• step6.txt the output file.
• step6_kappa.png a plot of the diffusivity kappa(x,y).
• step6_mesh.png a plot of the mesh.
• step6_u.png a plot of the solution.
• step6_u_grad_norm.png a plot of the norm of the gradient of the solution.