A FEniCS Tutorial - FEniCS Project

important extension of our first program is therefore to examine the computed
values of the solution, which is the focus of the present section.
A finite element function like u is expressed as a linear combination of basis
functions φ j , spanning the space V:
N∑
U j φ j . (13)
j=1
By writing solve(a == L, u, bc) in the program, a linear system will be
formed from a and L, and this system is solved for the U 1 ,...,U N values. The
U 1 ,...,U N values are known as degrees of freedom of u. For Lagrange elements
(and many other element types) U k is simply the value of u at the node with
global number k. (The nodes and cell vertices coincide for linear Lagrange
elements, while for higher-order elements there may be additional nodes at the
facets and in the interior of cells.)
Having u represented as a Function object, we can either evaluate u(x) at
any vertex x in the mesh, or we can grab all the values U j directly by
u_nodal_values = u.vector()
The result is a DOLFIN Vector object, which is basically an encapsulation
of the vector object used in the linear algebra package that is used to solve
the linear system arising from the variational problem. Since we program in
Python it is convenient to convert the Vector object to a standard numpy array
for further processing:
u_array = u_nodal_values.array()
With numpy arrays we can write ”MATLAB-like” code to analyze the data.
Indexing is done with square brackets: u_array[i], where the index i always
starts at 0.
Mesh information can be gathered from the mesh object, e.g.,
• mesh.coordinates() returns the coordinates of the vertices as an M ×d
numpy array, M being the number of vertices in the mesh and d being the
number of space dimensions,
• mesh.num_cells() returns the number of cells (triangles) in the mesh,
• mesh.num_vertices() returns the number of vertices in the mesh (with
our choice of linear Lagrange elements this equals the number of nodes),
Writing print mesh dumps a short, ”pretty print” description of the mesh
(print mesh actually displays the result of str(mesh)‘, which defines the pretty
print):
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