A FEniCS Tutorial - FEniCS Project

du = Function(V)
u = Function(V) # u = u_k + omega*du
omega = 1.0 # relaxation parameter
eps = 1.0
tol = 1.0E-5
iter = 0
maxiter = 25
while eps > tol and iter < maxiter:
iter += 1
A, b = assemble_system(a, L, bcs_du)
solve(A, du.vector(), b)
eps = numpy.linalg.norm(du.vector().array(), ord=numpy.Inf)
print ’Norm:’, eps
u.vector()[:] = u_k.vector() + omega*du.vector()
u_k.assign(u)
There are other ways of implementing the update of the solution as well:
u.assign(u_k) # u = u_k
u.vector().axpy(omega, du.vector())
# or
u.vector()[:] += omega*du.vector()
The axpy(a, y) operation adds a scalar a times a Vector y to a Vector object.
It is usually a fast operation calling up an optimized BLAS routine for the
calculation.
Mesh construction for a d-dimensional problem with arbitrary degree of the
Lagrange elements can be done as explained in Section 1.16. The complete
program appears in the file alg_newton_np.py.
2.3 A Newton Method at the PDE Level
Although Newton’s method in PDE problems is normally formulated at the
linear algebra level, i.e., as a solution method for systems of nonlinear algebraic
equations, we can also formulate the method at the PDE level. This approach
yields a linearization of the PDEs before they are discretized. FEniCS users will
probably find this technique simpler to apply than the more standard method
in Section 2.2.
Given an approximation to the solution field, u k , we seek a perturbation δu
so that
u k+1 = u k +δu (56)
fulfills the nonlinear PDE. However, the problem for δu is still nonlinear and
nothing is gained. The idea is therefore to assume that δu is sufficiently small
so that we can linearize the problem with respect to δu. Inserting u k+1 in the
PDE, linearizing the q term as
q(u k+1 ) = q(u k )+q ′ (u k )δu+O((δu) 2 ) ≈ q(u k )+q ′ (u k )δu, (57)
and dropping nonlinear terms in δu, we get
∇·(q(u k )∇u k) +∇·(q(u k )∇δu ) +∇·(q ′ (u k )δu∇u k) = 0.
54