Tangential tensors on immersed surfaces

[gobenavides]

Consider a immersed 2D manifold, e.g. mesh = IcosahedralSphereMesh(radius=1.0, refinement_level=3)

Is it possible to define a tensor-valued finite element space $V$ that is "tangential" from left and right. That is, if $P = I - n \otimes n$ is the projection onto the tangent plane, then for each $A \in V$ it holds that $PAP = A$?

I am particularly interested in the tangential version of the following

V = TensorFunctionSpace(mesh, "DG", 1)

[pbrubeck]

Possibly, but it might have to be symmetric

FunctionSpace(mesh, "Regge", 0)

Will give you a space of piecewise constants with tangential-tangential continuity. On a manifold the tensor should be tangential from left and right.

The broken version of this space might be closer to what you are after

FunctionSpace(mesh, BrokenElement(FiniteElement( "Regge", degree=0)))

[gobenavides]

Thanks for your answer. This may be useful for my needs.

In a similar vein, what about

FunctionSpace(mesh, "JM", 1)

which is also a space of symmetric tensors. In euclidean domains, this is Hdiv conforming. Is the manifold version implemented? Are their elements tangential from the left and right?

[pbrubeck]

JM is H(div) conforming, so the nn and nt components must be continuous along faces. Its columns and rows are mapped with the H(div) Piola transform. In order to get a tangential tensor from left and right, the columns and rows would need the H(curl) Piola map, which preserves the tangential trace.

JM is not currently implemented on manifolds, but we might try to implement them if they turn out to be interesting.

[pbrubeck]

The non-symmetric tangential DG0 element of your original question may be constructed as

element = WithMapping(TensorElement(FiniteElement("DG", degree=0), shape=(2, 2), "double covariant Piola")

FunctionSpace(mesh, element)

[gobenavides]

JM is H(div) conforming, so the nn and nt components must be continuous along faces. Its columns and rows are mapped with the H(div) Piola transform. In order to get a tangential tensor from left and right, the columns and rows would need the H(curl) Piola map, which preserves the tangential trace.

JM is not currently implemented on manifolds, but we might try to implement them if they turn out to be interesting.

Thanks, I didn't see this comment on time and ended up reporting this as an Issue. I apologize for that.

My collaborators Ricardo Nochetto and Johnny Guzman and I have some applications in mind for JM on manifolds. We're looking forward for this new implementation! Thanks in advance.

[gobenavides]

@pbrubeck By the way, are there any available finite elements in Firedrake that are Hdiv conforming on manifolds and tangential from left and right? They don't need to be symmetric. Thanks!

[pbrubeck]

@pbrubeck By the way, are there any available finite elements in Firedrake that are Hdiv conforming on manifolds and tangential from left and right? They don't need to be symmetric. Thanks!

I believe these two requirements contradict each other. Tangential elements would map with the covariant Piola transform. H(div) conforming elements map with contravariant Piola transform. Do you agree?