How can I write this diffusion problem variational formulation correctly?

variational formulation
1

Hi ! I’m a beginner and I wanted to write a simple code but it doesn’t work… Can anyone help me ?

base = Rectangle(Point(-1.0, -1.0), Point(1.0, 1.0))
hole = Circle(Point(0.25, 0), 0.25)
mesh5 = generate_mesh(base - hole, 15)

plt.figure(1)
plot(mesh5, title="Maillage du rectangle troué")

Hh5 = FunctionSpace(mesh5, 'P', 1)
tol = 1E-14
class inneredge(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (abs(x[0]-0.25)<0.25+tol) and (abs(x[1])<0.25+tol) 
    
ie = inneredge()
ie = DirichletBC(Hh5, Constant(1.0), ie)

class outeredge(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and (abs(x[0] - 0.25) > 0.25+tol) and (abs(x[1])>0.25+tol)
    
oe = inneredge()
oe = DirichletBC(Hh5, Constant(0.0), oe)

c=Expression(("(x[0]-0.5)*(x[0]-0.5)+x[1]*x[1]","0.0"),degree = 2)

u = TrialFunction(Hh5)
v = TestFunction(Hh5)

F = inner(grad(u), grad(v))*dx+dot(c,grad(u))*v*dx

u = Function(Hh5)

solve(F == 0, u, [ie,oe],solver_parameters={"newton_solver":
                                            {"relative_tolerance": 1e-6}})

plu=plot(u5)
plt.title('Solution numérique u5')
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar(plu)

Here is my error
*** Error: Unable to define nonlinear variational problem F(u; v) = 0 for all v.
*** Reason: Expecting the residual F to be a linear form (not rank 2).
*** Where: This error was encountered inside NonlinearVariationalProblem.cpp.

2

You should use u=Function(V), not TrialFunction(V) if you want to solve this as a non-linear problem.
However, as your problem is not non-linear, you could keep the trial function and use:

a = lhs(F)
L = rhs(F)
uh = Function(Hh5)
solve(a==L, uh, [ie,oe])
3

Thank you very much it’s working put the output seems to haven’t taken in account the boundary conditions, do you know why ?