A FEniCS Tutorial - FEniCS Project

problem = NonlinearVariationalProblem(F, u, bcs, J)
solver = NonlinearVariationalSolver(problem)
solver.solve()
where F corresponds to the nonlinear form F(u;v), u is the unknown Function
object, bcs represents the essential boundary conditions (in general a list of
DirichletBC objects), and J is a variational form for the Jacobian of F.
Let us explain in detail how to use the built-in tools for nonlinear variational
problems and their solution. The F form corresponding to (41) is straightforwardly
defined as follows, assuming q(u) is coded as a Python function:
u_ = Function(V) # most recently computed solution
v = TestFunction(V)
F = inner(q(u_)*nabla_grad(u_), nabla_grad(v))*dx
Note here that u_ is a Function (not a TrialFunction). An alternative and
perhaps more intuitive formula for F is to define F(u;v) directly in terms of a
trial function for u and a test function for v, and then create the proper F by
u = TrialFunction(V)
v = TestFunction(V)
F = inner(q(u)*nabla_grad(u), nabla_grad(v))*dx
u_ = Function(V) # the most recently computed solution
F = action(F, u_)
The latter statement is equivalent to F(u = u − ;v), where u − is an existing
finite element function representing the most recently computed approximation
to the solution. (Note that u k and u k+1 in the previous notation correspond to
u − and u in the present notation. We have changed notation to better align the
mathematics with the associated UFL code.)
ThederivativeJ (J)ofF (F)isformallytheGateauxderivativeDF(u k ;δu,v)
of F(u;v) at u = u − in the direction of δu. Technically, this Gateaux derivative
is derived by computing
lim
ǫ→0
d
dǫ F i(u − +ǫδu;v). (62)
The δu is now the trial function and u − is the previous approximation to the
solution u. We start with
∫
d
∇v ·(q(u − +ǫδu)∇(u − +ǫδu)) dx
dǫ
Ω
and obtain
∫
∇v ·[q ′ (u − +ǫδu)δu∇(u − +ǫδu)+q(u − +ǫδu)∇δu] dx,
Ω
which leads to
∫
Ω
∇v ·[q ′ (u − )δu∇(u − )+q(u − )∇δu] dx, (63)
56