FE basis transformations

Notation

Consider a reference polytope $\hat{K}$, mapped to the physical space by a geometrical map $F$, i.e. $K = F(\hat{K})$. Consider also a linear function space on the reference polytope $\hat{V}$, and a set of unisolvent degrees of freedom represented by moments in the dual space $\hat{V}_*$.

Throughout this document, we will use the following notation:

• $\varphi \in V$ is a physical field $\varphi : K \rightarrow \mathbb{R}^k$. A basis of $V$ is denoted by $\Phi = \{\varphi\}$.
• $\hat{\varphi} \in \hat{V}$ is a reference field $\hat{\varphi} : \hat{K} \rightarrow \mathbb{R}^k$. A basis of $\hat{V}$ is denoted by $\hat{\Phi} = \{\hat{\varphi}\}$.
• $\sigma \in V^*$ is a physical moment $\sigma : V \rightarrow \mathbb{R}$. A basis of $V^*$ is denoted by $\Sigma = \{\sigma\}$.
• $\hat{\sigma} \in \hat{V}_*$ is a reference moment $\hat{\sigma} : \hat{V} \rightarrow \mathbb{R}$. A basis of $\hat{V}_*$ is denoted by $\hat{\Sigma} = \{\hat{\sigma}\}$.

Pullbacks and Pushforwards

We define a pushforward map as $F_* : \hat{V} \rightarrow V$, mapping reference fields to physical fields. Given a pushforward $F_*$, we define:

• The pullback $F^* : V^* \rightarrow \hat{V}^*$, mapping physical moments to reference moments. Its action on physical dofs is defined in terms of the pushforward map $F_*$ as $\hat{\sigma} = F^*(\sigma) := \sigma \circ F_*$.
• The inverse pushforward $(F_*)^{-1} = F_{-*} : V \rightarrow \hat{V}$, mapping physical fields to reference fields.
• The inverse pullback $(F^*)^{-1} = F^{-*} : \hat{V}^* \rightarrow V^*$, mapping reference moments to physical moments. Its action on reference dofs is defined in terms of the inverse pushforward map $F_{-*}$ as $\sigma = F^{-*}(\hat{\sigma}) := \hat{\sigma} \circ F_{-*}$.

Change of basis

In many occasions, we will have that (as a basis)

\[\hat{\Sigma} \neq F^*(\Sigma), \quad \text{and} \quad \Phi \neq F_*(\hat{\Phi})\]

To maintain conformity and proper scaling in these cases, we define cell-dependent invertible changes of basis $P$ and $M$, such that

\[\hat{\Sigma} = P F^*(\Sigma), \quad \text{and} \quad \Phi = M F_*(\hat{\Phi})\]

An important result from [1, Theorem 3.1] is that $P = M^T$.

We then have the following diagram:

\[\hat{V}^* \xleftarrow{P} \hat{V}^* \xleftarrow{F^*} V^* \\ \hat{V} \xrightarrow{F_*} V \xrightarrow{P^T} V\]

In order to map a reference FE to a physical FE, $\Sigma$ and $\Phi$ are obtained by mapping $\hat\Sigma$ and $\hat\Phi$ using

\[\Phi = M F_*(\hat{\Phi}), \quad \text{and} \quad \Sigma = M^{-T} F^{-*}(\hat\Sigma).\]

In the implementation, $M$ is replaced with $M'=DM$ (and $M^{-T}$ by $D^{-T}M^{-T}$). $D$ is an optional diagonal matrix used to rescale the DOFs that would otherwise scale with the local mesh-size $h$. $M$ takes into account sign changes due to orientation change (see e.g. NormalSignMap) and non-trivial relation between the physical and reference FE (see non affine-equivalent FEs in [1]-[2]).

$M$ and its transposed inverse are computed cell wise by compute_cell_bases_changes. From an implementation point of view, it is more natural to build $M^{-T}$ and then retrieve $M$ by transposition and inversion. $D$'s entries are computed using the scaling function defined by get_dofscale_setter_function. $\Sigma$ and $\Phi$ are finally put together in get_cell_shapefuns_and_dof_basis.

Interpolation

In each cell $K$ and for $C_b^k(K)$ the space of functions defined on $K$ with at least $k$ bounded derivatives, we define the interpolation operator $I_K : C_b^k(K) \rightarrow V$ as

\[I_K(g) = \Sigma(g) \Phi \quad, \quad \Sigma(g) = M^{-T} \hat{\Sigma}(F_{-*}(g))\]

Implementation notes

References

[1] Kirby 2017, A general approach to transforming finite elements.

[2] Aznaran et al. 2021, Transformations for Piola-mapped elements.