Hi, I intend to implement a system of PDEs (4‍‍×4) in Firedeake package, which the function spaces are mix of tensor, vector and scalar. For this reason I prefer to understand the code of mixed formulation for Poisson equation completly.

In the mixed formulation code of Poisson equation, when I change the function space from "FunctionSpace(mesh, "BDM", 1)" to "VectorFunctionSpace(mesh, "CG", 1)", there is a bug in solvig the system. I implement the following code:

mesh = UnitSquareMesh(10, 10)

VCG = VectorFunctionSpace(mesh, "CG", 1)

DG = FunctionSpace(mesh, "DG", 0)
W = VCG * DG

sigma, u = TrialFunctions(W)
tau, v = TestFunctions(W)
x, y = SpatialCoordinate(mesh)
f = Function(DG).interpolate(
    10*exp(-(pow(x - 0.5, 2) + pow(y - 0.5, 2)) / 0.02))

a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx

bc0 = DirichletBC(W.sub(0), as_vector([0.0, -sin(5*x)]), 1)
bc1 = DirichletBC(W.sub(0), as_vector([0.0, sin(5*y)]), 2)

w = Function(W)

solve(a == L, w, bcs=[bc0, bc1])
sigma, u = w.split()

File("poisson_mixed.pvd").write(u)

try:
  import matplotlib.pyplot as plt
except:
  warning("Matplotlib not imported")

try:
  plot(u)
except Exception as e:
  warning("Cannot plot figure. Error msg '%s'" % e)

try:
  plt.show()
except Exception as e:
  warning("Cannot show figure. Error msg '%s'" % e)

f.interpolate(cos(x*pi*2)*cos(y*pi*2))
print(sqrt(assemble(dot(u - f, u - f) * dx)))

and I get the following error:

Traceback (most recent call last):
  File "test2.py", line 156, in <module>
    solve(a == L, w, bcs=[bc0, bc1])
  File "/home/amireh/firedrake/src/firedrake/firedrake/solving.py", line 121, in solve
    _solve_varproblem(*args, **kwargs)
  File "/home/amireh/firedrake/src/firedrake/firedrake/solving.py", line 151, in _solve_varproblem
    solver.solve()
  File "/home/amireh/firedrake/src/firedrake/firedrake/variational_solver.py", line 242, in solve
    solving_utils.check_snes_convergence(self.snes)
  File "/home/amireh/firedrake/src/firedrake/firedrake/solving_utils.py", line 231, in check_snes_convergence
    %s""" % (snes.getIterationNumber(), msg))
firedrake.exceptions.ConvergenceError: Nonlinear solve failed to converge after 0 nonlinear iterations.
Reason:
   Inner linear solve failed to converge after 10000 iterations with reason: DIVERGED_MAX_IT