A FEniCS Tutorial - FEniCS Project

A = assemble(a, exterior_facet_domains=boundary_parts)
b = assemble(L, exterior_facet_domains=boundary_parts)
Then essential boundary conditions are enforced, and the system can be solved
in the usual way:
for bc in bcs:
bc.apply(A, b)
u = Function(V)
U = u.vector()
solve(A, U, b)
The complete code is in the dnr_p2D.py file in the stationary/poisson directory.
6 More Examples
Many more topics could be treated in a FEniCS tutorial, e.g., how to solve systems
of PDEs, how to work with mixed finite element methods, how to create
more complicated meshes and mark boundaries, and how to create more advanced
visualizations. However, to limit the size of this tutorial, the examples
end here. There are, fortunately, a rich set of FEniCS demos. The FEniCS documentation
explains a collection of PDE solvers in detail: the Poisson equation,
the mixed formulation for the Poission equation, the Biharmonic equation, the
equations of hyperelasticity, the Cahn-Hilliard equation, and the incompressible
Navier-Stokes equations. Both Python and C++ versions of these solvers
are explained. An eigenvalue solver is also documented. In the dolfin/demo
directory of the DOLFIN source code tree you can find programs for these and
many other examples, including the advection-diffusion equation, the equations
of elastodynamics, a reaction-diffusion equation, various finite element methods
for the Stokes problem, discontinuous Galerkin methods for the Poisson and
advection-diffusion equations, and an eigenvalue problem arising from electromagnetic
waveguide problem with Nedelec elements. There are also numerous
demos on how to apply various functionality in FEniCS, e.g., mesh refinement
and error control, moving meshes (for ALE methods), computing functionals
over subsets of the mesh (such as lift and drag on bodies in flow), and creating
separate subdomain meshes from a parent mesh.
Theprojectcbc.solve(https://launchpad.net/cbc.solve)offersmorecomplete
PDE solvers for the Navier-Stokes equations, the equations of hyperelasticity,
fluid-structure interaction, viscous mantle flow, and the bidomain
model of electrophysiology. Most of these solvers are described in the ”FEniCS
book”[14](https://launchpad.net/fenics-book). Anotherproject,cbc.rans
(https://launchpad.net/cbc.rans), offers an environment for very flexible
and easy implementation of Navier-Stokes solvers and turbulence [20, 19]. For
example, cbc.rans contains an elliptic relaxation model for turbulent flow involving
18 nonlinear PDEs. FEniCS proved to be an ideal environment for
implementing such complicated PDE models. The easy construction of systems
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