A FEniCS Tutorial - FEniCS Project

which is nothing but a linear system,
AU = b,
where the entries in A and b are given by
A ij = a(φ j , ˆφ i ),
b i = L(ˆφ i ).
The examples so far have specified the left- and right-hand side of the variational
formulation and then asked FEniCS to assemble the linear system and
solve it. An alternative to is explicitly call functions for assembling the coefficient
matrix A and the right-side vector b, and then solve the linear system
AU = b with respect to the U vector. Instead of solve(a == L, u, bc) we
now write
A = assemble(a)
b = assemble(L)
bc.apply(A, b)
u = Function(V)
U = u.vector()
solve(A, U, b)
The variables a and L are as before. That is, a refers to the bilinear form
involving a TrialFunction object (say u) and a TestFunction object (v), and
L involves a TestFunction object (v). From a and L, the assemble function
can compute A and b.
The matrix A and vector b are first assembled without incorporating essential
(Dirichlet) boundary conditions. Thereafter, the call bc.apply(A, b)
performs the necessary modifications of the linear system such that u is guaranteed
to equal the prescribed boundary values. When we have multiple Dirichlet
conditions stored in a list bcs, as explained in Section 1.14, we must apply each
condition in bcs to the system:
# bcs is a list of DirichletBC objects
for bc in bcs:
bc.apply(A, b)
There is an alternative function assemble_system, which can assemble the
system and take boundary conditions into account in one call:
A, b = assemble_system(a, L, bcs)
The assemble_system function incorporates the boundary conditions in the
element matrices and vectors, prior to assembly. The conditions are also incorporated
in a symmetric way to preserve eventual symmetry of the coefficient
matrix. Withbc.apply(A, b) the matrix A is modified in an unsymmetric way.
Note that the solution u is, as before, a Function object. The degrees of
freedom, U = A −1 b, are filled into ‘u‘’s Vector object (u.vector()) by the
solve function.
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