Projection of fine mesh solution to coarse mesh in order to use it for error estimates.

[MariosChristop]

Hello,

I am trying to project a fine mesh solution to a coarse mesh in order to use it for error estimates. By reading the documentation, specifically firedrake/projection.py I found out two ways of doing this. One is by using the firedrake.project() and the other is by using firedrake.Projector(). The problem is that I am getting different error estimates when I am trying to calculate the error norms in the same problem. What is the difference between the two methods?

The problem I am trying to solve is the stationary Convection-Diffusion

$$ -10^{-4} \Delta u + u_{x} + u_{y} = 1, \text{ in } \Omega=(0,1)^{2}$$ $$ u=0, \text{ on } \partial \Omega $$

Below I provide the code for the fine mesh solution.

# # Make 2D rectangle from (-1, -1) to (0, 1)
rect2 = WorkPlane(Axes((0,0,0), n=Z, h=X)).Rectangle(1,1).Face()
L = rect2
geo = OCCGeometry(L, dim=2)
ngmesh = geo.GenerateMesh(maxh=0.001)
Fine_mesh = Mesh(ngmesh)
VTKFile("Lshape/lshaped.pvd").write(Fine_mesh)

(x, y) = SpatialCoordinate(Fine_mesh)

f_fine = Constant(1)
c_fine = Constant(1)
epsilon_fine=Constant(10e-4)

beta_fine = as_vector((1.0,1.0))

V_fine = FunctionSpace(Fine_mesh, "Lagrange", 1)

u_fine = TrialFunction(V_fine)
v_fine = TestFunction(V_fine)

A_CV_fine = (epsilon_fine*dot(grad(u_fine), grad(v_fine))+ dot(dot(beta_fine, grad(u_fine)), v_fine))*dx 

L_fine = dot(f_fine, v_fine)*dx

bc_ref_fine = DirichletBC(V_fine, 0, "on_boundary")

uh_fine = Function(V_fine)
solve(A_CV_fine==L_fine, uh_fine, bc_ref_fine)

After I do a similar thing in order to obtain the coarse mesh solution I am using the code below for the projection

# New functions to assign the projection
u_new_firedrake = firedrake.Function(V) # V is the Function space obtain by the coarse mesh
u_new_firedrake_2 = firedrake.Function(V)

#1st Method
firedrake.Projector(uh_fine, u_new_firedrake)

#2nd Method
firedrake.project(uh_fine, u_new_firedrake_2)

and I get the following results

errornorm(uh_coarse, u_new_firedrake, norm_type="L2")

>np.float64(0.4079244605162776)

errornorm(uh_coarse, u_new_firedrake_2, norm_type="L2")

>np.float64(0.03887241581016237)

Also when I use the two methods mentioned above I don't enforce the boudnary conditions because i get this:

firedrake.Projector(uh_fine, u_new_firedrake, bc_ref)

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
[/tmp/ipython-input-34-1168401598.py](https://localhost:8080/#) in <cell line: 0>()
----> 1 firedrake.Projector(uh_fine, u_new_firedrake, bc_ref)

petsc4py/PETSc/Log.pyx in petsc4py.PETSc.Log.EventDecorator.decorator.wrapped_func()

petsc4py/PETSc/Log.pyx in petsc4py.PETSc.Log.EventDecorator.decorator.wrapped_func()

[/usr/local/lib/python3.11/dist-packages/firedrake/projection.py](https://localhost:8080/#) in Projector(v, v_out, bcs, solver_parameters, form_compiler_parameters, constant_jacobian, use_slate_for_inverse, quadrature_degree)
    357     else:
    358         if bcs is not None:
--> 359             raise ValueError("Haven't implemented supermesh projection with boundary conditions yet, sorry!")
    360         if not isinstance(source, function.Function) or source.ufl_element().family() == "Real":
    361             raise NotImplementedError("Only for source Functions, not %s" % type(source))

ValueError: Haven't implemented supermesh projection with boundary conditions yet, sorry!

Does this mean that my projection does not 'obey' the boundary conditions? Thank you for taking the time to answer my queastion.

[connorjward]

I'm pretty sure you're getting different results because you need to call Projector.project to perform the actual computation. This is exactly what the project free function does so you should get the same answer.

And regarding the boundary condition error you're unfortunately right. We don't enforce boundary conditions for projection where there are multiple meshes.