Hello everyone,

I am interested in a mixed formulation where the test and trial spaces are defined on different (but nested) meshes. Is this somehow possible in FEniCSx?

Thanks!

Hello everyone,

I am interested in a mixed formulation where the test and trial spaces are defined on different (but nested) meshes. Is this somehow possible in FEniCSx?

Thanks!

I am adding a MWE that tries to implement it.

```
from dolfinx import mesh, fem
import ufl
from mpi4py import MPI
msh_trial = mesh.create_unit_square(MPI.COMM_WORLD, 5, 5, mesh.CellType.quadrilateral)
msh_test = mesh.create_unit_square(MPI.COMM_WORLD, 10, 10, mesh.CellType.quadrilateral)
U = fem.functionspace(msh_trial, ("Lagrange", 2))
V = fem.functionspace(msh_test, ("Lagrange", 1))
u = ufl.TrialFunction(U)
v = ufl.TestFunction(V)
dx = ufl.Measure("dx", domain=msh_trial)
a = ufl.inner(u, v) * dx
A = fem.assemble_matrix(fem.form(a))
```

As it could be expected, the problem is in the meshes, as I get the following error.

```
RuntimeError: Incompatible mesh. entity_maps must be provided.
```

Note that I still get the same error with `msh_trial = mesh.create_unit_square(MPI.COMM_WORLD, 10, 10, mesh.CellType.quadrilateral)`

.

Hello Ivan,

currently `dolfinx`

does not allow for this use case. Internally it is assumed that a variational form lives on a single mesh and thus integration only supports assembly with different function spaces defined on the same mesh.

However, there is some recent work regarding the interaction of meshes that are produced by refinement operations - in the context of geometric multigrid. This should (in the future - see this PR) allow for the transfer of functions between the two spaces you proposed. Note however that there is currently no quadrilateral refinement routine available, so this requires the usage of triangles/tetrahedrons.

In general the code of how such a matrix between two different spaces (with different meshes) may be constructed can be found here. Note: currently this only supports P^1 elements and needs to be extended to general dof maps in the future.

Another thought: Maybe introducing an auxiliary space P^2 on the finer mesh as an ‘integration space’ (for at least a qualitative analysis) could be worth considering.

Hope that helps,

Paul

1 Like

Good morning folks,

Just in sake of curiosity, what are the physical problem that require a FE method where the trial and test function are not defined on the same space/mesh ?

Regards,

Pierre.

I know some mixed dimensional methods with non-conforming grids where this could be required, ref: https://arxiv.org/pdf/2312.16565

Another possible application I’m aware of is with partition of unity methods such as XFEM or GFEM. Here the approximation space is constructed by connecting local approximation spaces (FEM spaces themselves) \{V_i \}_{i\in I} by a partition of unity \{ \phi_i \}_{i \in I} (most commonly this is just given by P^1 elements) to form an approximation space \{ \phi_i V_i \}_{i \in I}. In this setting it is often advantageous to be able to control if the PU is considered in the testing and trial arguments. This results in more control over runtime (not using the PU is faster to integrate/assemble) and allows for great preconditioning opportunities which make use of the localized approximation space structure.

Equal ideas can be employed by changing the quadrature rules used for the assembly of forms with classic FEM, which may be facilitated by using a different space/mesh.