introduce MixedMesh class

[ksagiyam]

First half of #264

Introduce MixedMesh class for multi-mesh problems.

Note that extract_domains() now uses more robust sort_domains() inside, so it might behave slightly differently.

Edit 12-09-2024:

MixedMesh class represents a collection of meshes (e.g., submeshes) that, along with a MixedElement, can represent a mixed function space defined across multiple domains.
The motivation is to allow for treating Arguments and Coefficients on a mixed function space defined across multiple domains just like those on a mixed function space defined on a single mesh.

Specifically, the following becomes possible (see tests for more):

    cell = triangle
    mesh0 = Mesh(FiniteElement("Lagrange", cell, 1, (2,), identity_pullback, H1))
    mesh1 = Mesh(FiniteElement("Lagrange", cell, 1, (2,), identity_pullback, H1))
    domain = MixedMesh([mesh0, mesh1])
    elem0 = FiniteElement("Lagrange", cell, 1, (), identity_pullback, H1)
    elem1 = FiniteElement("Lagrange", cell, 2, (), identity_pullback, H1)
    elem = MixedElement([elem0, elem1])
    V = FunctionSpace(domain, elem)
    v = TestFunction(V)
    v0, v1 = split(v)

For now, one can only perform cell integrations when MixedMeshes are used. This is because, e.g., an interior facet integration on a submesh may either be interior or exterior facet integration on the parent mesh, and we need a logic to apply default restrictions on coefficients defined on the participating meshes. This is the second half of #264.

Also, currently, all component meshes must have the same cell type (and thus the same topological dimension) -- we are to remove this limitation in the future.

Core changes:

• GradRuleSet.{reference_value, reference_grad} work component-wise (component of the mixed space) if the FunctionSpace is defined on a MixedMesh, so that each component is associated with a component of the MixedMesh, say domain_i (JacobianInverse(domain_i) is then well defined).
• extract_arguments_and_coefficients is now called extract_terminals_with_domain, and it now also collects GeometricQuantys, so that we can correctly handle, say SpatialCoordinate(mesh1) and SpatialCoordinate(mesh2), in problem solving environments.

[ksagiyam]

@jpdean @jorgensd This is what we briefly discussed at PDESoft. Could you have a look?

[jorgensd]

Could we just call this list base_types, as it is quite explicit from the function call what it does, and it makes the latter code more readable.

[ksagiyam]

Renamed it terminals.

[jorgensd]

What else com gq._domain be? is this relying on the deprecated definition that a domain can be defined just through a ufl.Cell?

[ksagiyam]

No. It checks if gq._domain is Mesh and not MixedMesh.

[jorgensd]

So we now have two ways of creating a FunctionSpace with MixedElements.

1. A single mesh of class Mesh with a MixedElement, where MixedElement can have arbitrary many elements
2. A MixedMesh consisting of M meshes, and a MixedElement consisting of M elements.

Do we need additional checks in the FunctionSpace constructor to ensure that this is satisfied (instead of catching this at a later instance, inside for instance GenericDerivativeRuleset.reference_value.

I would also like to highlight that the API for MixedElement and MixedMeshes differs, as one passes a list, while the other unrolls a list, I think we should be consistent, and use lists in both places

[ksagiyam]

Added checks in FunctionSpace constructor and changed MixedMesh API as suggested. Thanks.

[jorgensd]

Could we give this a more obvious name?
Isn't this isn't this just the ufl_shape of the input operand, which should always be the same as the ufl_shape of o.
Another question (as I don't quite see how this works) is what is the ref_dim?, which would be the topological dimension of the unique "ufl-domain" of o. But what is the topological dimension of a MixedMesh?

Does this mean that a MixedMesh cannot have co-dimension 1 meshes in them, i.e. a Mesh of triangles and a Mesh of intervals?

[ksagiyam]

Turned out vsh could be removed, so I did.

You are right. Currently, MixedMesh can only be composed of meshes of the same cell_type, so topological dimension is well defined. I edited the doc string of MixedMesh to make this point clear.

[garth-wells]

It would be helpful to have a PR summary, e.g. what it sets out the achieve, what it supports, designs considered, design rationale, implementation summary, and what a MixedMesh is and how a MixedMesh is different from a Mesh.

[ksagiyam]

Updated PR summary.

Added more checks.

Fixed MixedPullback.physical_value_shape().

[ksagiyam]

Can I have another round of review on this?