Why is the error of gradient of solution of Poisson/0-Hodge-Laplace equation so large?

[Alpha-Legion]

I solved a Poisson equation on a unit sphere. The codes are as follows:
'''

It solves 0-Hodge-Laplace equation on surfaces.

from firedrake import *

mesh & coordinates

mesh = UnitIcosahedralSphereMesh(refinement_level=5)
mesh.init_cell_orientations(SpatialCoordinate(mesh))
x, y, z = SpatialCoordinate(mesh)

functions

F = FunctionSpace(mesh, "CG", 1)
V = VectorFunctionSpace(mesh, "CG", 1)

u = TrialFunction(F)
v = TestFunction(F)

source term

f = Function(F)
f.interpolate(-2*z)

variational formulation

a = (inner(grad(u), grad(v))) * dx
L = f * v * dx

solving system

nullspace = VectorSpaceBasis(constant=True)
u = Function(F)
solve(a == L, u, nullspace=nullspace)

true solution & error

u_ = Function(F)
u_.interpolate(-z)

error = sqrt(assemble(dot(u - u_, u - u_) * dx))/sqrt(assemble(dot(u_, u_) * dx))
print(error)

du = Function(V)
du.interpolate(grad(u))
du_ = Function(V)
du_.assign(project(as_vector([zx, zy, -sqrt(xx + yy)]), V))
error = sqrt(assemble(dot(du - du_, du - du_) * dx))/sqrt(assemble(dot(du_, du_) * dx))
print(error)
'''
The output errors for the solution and its gradient are:
0.00036147241076426006
0.1658159619652392
respectively.

The error of the solution is good. But the error of the gradient is not as small as expected. What's wrong?
Thanks!

[Alpha-Legion]

Sorry, the true solution is wrong....