Construction of FunctionSpace

1

Hello everyone. I have the following spaces:
V1 = \left\{v \in L^{2}(\Omega)^{2} \mbox{such that} v|_{T}\in P^{k}(T)\times P^{k}(T) + xP^{k}(T) \forall T\in \mathcal{T}_{h}\right\}
V2 = \left\{w \in L^{2}(\mathcal{E}_{i,h}) \mbox{such that} w|_{e} \in P^{k}(e) \forall e \in \mathcal{E}_{i,h}\right\}
I am not able to understand how can I define these in FeniCS. For V1 , if I use Raviart Thomas element then with that we have continuity of the normal components. But I do not have that.
For V2,
V2 = FunctionSpace ( mesh, ā€œDGā€, 1 , restriction = ā€˜facetā€™)
is this enough?
I do not understand anything about V1, how can I construct it?

Thank you in advance.

2

Do you have a reference for the space V1, specifically which basis functions you expect to use for this space. Itā€™s not a space Iā€™m currently familiar with

3

Thank you for your reply. In the following paper, the above space is specified. Kindly look equation (2.6)-(2.8). In these equations, the above spaces are used.
https://sci-hub.se/10.1137/S0036142902417893

Thank you in advance.

4

Thanks. It looks like the space V_h in that reference is Raviart-Thomas (see DefElement: Raviartā€“Thomas), so you can create it with:

V1 = FunctionSpace(mesh, "Raviart-Thomas", 1)
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