Solve Poisson equation on surfaces

[Alpha-Legion]

I tried to solve an inhomogenous Beltrami-Laplace equation on the unit sphere. The codes are as follows:

from firedrake import *

mesh = UnitIcosahedralSphereMesh(refinement_level=3)
mesh.init_cell_orientations(SpatialCoordinate(mesh))

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

u = TrialFunction(V)
v = TestFunction(V)

f = Function(V)
x, y, z = SpatialCoordinate(mesh)
f.interpolate(x*y*z)

a = (inner(grad(u), grad(v))) * dx
L = f * v * dx
nullspace = VectorSpaceBasis(constant=True)

u = Function(V)

solve(a == L, u, nullspace=nullspace)

Question: Is this the correct formulation? Especially, is the non-Euclidean metric taken cared of automatically? Equivalently, are the operators defined independent of coordinate changes?

[Alpha-Legion]

Yes. I find the answer.