Hi, I am learning about fenics and I started with examples with TrialFunctions and TestFunctions. But after a few examples, I found most example doesn’t bother to define a trial function.
Is it the same using function instead of trialfunction?
Regards
Juntao
TrialFunctions are needed if you want to define a linear variational problem directly as a(u,v)=L(v) and call solve(a==L,...). Here, u is a TrialFunction and v a TestFunction.
In the case you define the variational problem as F(u,v)=0 and call solve(F==0, ...) the problem is treated as nonlinear and u is a Function and v is a TestFunction.
The reason for these different definitions, however, is unclear to me.
Consider the following Poisson example where I demonstrate both ways of defining the same problem:
mesh = UnitIntervalMesh(8)
V = FunctionSpace(mesh, "Lagrange",1)
# example RHS
f = Expression("1.-pow(x[0]-.5,2)", element = V.ufl_element())
### Define as a(u,v) = L(v)
u = TrialFunction(V)
v = TestFunction(V)
# define bilinear and linear forms directly
a = inner(grad(u),grad(v))*dx
L = f*v*dx
# define function for storing solution
sol = Function(V)
solve(a==L, sol, bcs = DirichletBC(V, Constant(0.),lambda x,on_boundary: on_boundary))
### Define as form F(u,v) = 0
u = Function(V)
v = TestFunction(V)
F = inner(grad(u), grad(v))*dx - f*v*dx
solve(F==0, u, bcs = DirichletBC(V, Constant(0.),lambda x,on_boundary: on_boundary))
Hi, klunkean,
Thank you so much. I would never be able to figure this out myself. Though I’m still confused about the subtle difference. Does F==0 implies a non-linear solving, thus a function is used instead of trial function? I don’t know. But still, thanks a lot.
Yes, if you call solve(F==0, ...) a NonlinearVariationalProblem is created in the background and solved by Newton iteration. If the problem F(u,v)=0 is linear, however, the solver will converge in one step.
Is not totally related to your question but consider that u=Function(V) mathematically produces
u=\sum_i u_i \phi_i
That is a function constructed by the basis function of the Space defined in the galerkin method