A FEniCS Tutorial - FEniCS Project

FEniCS is quite limited. And Python is an easy-to-learn language that you
certainly will love and use far beyond FEniCS programming.) Section 7.9 lists
some relevant Python books.
The listed FEniCS program defines a finite element mesh, the discrete function
spaces V and ˆV correspondingto this mesh and the element type, boundary
conditions for u (the function u 0 ), a(u,v), and L(v). Thereafter, the unknown
trial function u is computed. Then we can investigate u visually or analyze the
computed values.
The first line in the program,
from dolfin import *
imports the key classes UnitSquare, FunctionSpace, Function, and so forth,
from the DOLFIN library. All FEniCS programs for solving PDEs by the finite
element method normally start with this line. DOLFIN is a software library
with efficient and convenient C++ classes for finite element computing, and
dolfin is a Python package providing access to this C++ library from Python
programs. You can think of FEniCS as an umbrella, or project name, for a
set of computational components, where DOLFIN is one important component
for writing finite element programs. The from dolfin import * statement
imports other components too, but newcomers to FEniCS programming do not
need to care about this.
The statement
mesh = UnitSquare(6, 4)
defines a uniform finite element mesh over the unit square [0,1] × [0,1]. The
mesh consists of cells, which are triangles with straight sides. The parameters
6 and 4 tell that the square is first divided into 6 × 4 rectangles, and then
each rectangle is divided into two triangles. The total number of triangles then
becomes 48. The total number of vertices in this mesh is 7·5 = 35. DOLFIN
offers some classes for creating meshes over very simple geometries. For domains
of more complicated shape one needs to use a separate preprocessor program to
create the mesh. The FEniCS program will then read the mesh from file.
Having a mesh, we can define a discrete function space V over this mesh:
V = FunctionSpace(mesh, ’Lagrange’, 1)
The second argument reflects the type of element, while the third argument is
the degree of the basis functions on the element. The type of element is here
”Lagrange”, implying thestandard Lagrangefamily of elements. (Some FEniCS
programs use ’CG’, for Continuous Galerkin, as a synonym for ’Lagrange’.)
With degree 1, we simply get the standard linear Lagrange element, which is a
triangle with nodes at the three vertices. Some finite element practitioners refer
to this element as the ”linear triangle”. The computed u will be continuous
and linearly varying in x and y over each cell in the mesh. Higher-degree
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