SymPy for preprocessing examples

[gdmcbain]

While dependence on SymPy was dropped in #37 for computing basis functions, it might be useful for preprocessing certain examples. It would not then be a dependence for the scikit-fem package, just for the examples.

One is suggested by the FEniCS tutorial example ft05_nonlinear_poisson.py. The example is basically like docs/examples/ex10.py, except that it adds an artificial right-hand side to balance the equation for a specified solution; i.e. it uses the method of manufactured solutions. That right-hand side is computed from the stipulated exact solution using SymPy.

A translation of the FEniCS example into scikit-fem is at https://github.com/gdmcbain/fenics-tuto-in-skfem/tree/master/05_poisson_nonlinear. It goes beyond the original in using:

• sympy.vector for gradient and divergence
• sympy.utilities.lambdify to evaluate symbolic vector fields on points specified by a numpy.ndarray like Mesh.p

There are a few examples (ex12, ex13, ex14, ex16, ex17, ex18, ex19, ex20) which involve exact solutions; I wonder whether it would be worth modifying one or more of them to use SymPy to compute the exact solution to show how it's done here.

[gdmcbain]

According to Cellier & Ruyer-Quil (2019), Scikit-FDiff uses SymPy to allow ‘easy and automated finite difference discretization’; might be worth a look for inspiration or comparison.

• Cellier, N. & Ruyer-Quil, C. (2019). scikit-finite-diff, a new tool for PDE solving. Journal of Open Source Software, 4, 1356. doi:10.21105/joss.01356