In a lot of papers discussing discontinuous Galerkin methods for solving elliptic/parabolic PDE's, they mention a "lift" operator.
For example, consider a simple Poisson system: \begin{gather} \nabla \cdot \mathbf{\sigma} = f\\ \mathbf{\sigma} = \nabla u \end{gather} with an appropriate choice of boundary conditions (not of concern here).
In Arnold et al.'s paper, they define a lift operator as such:
Suppose you have two non-overlapping subsets of a domain $\Omega$ denoted as the $K_0$ and the $K_1$ subsets such that $\Omega = K_0 \cup K_1$, and a shared interface $\Gamma = K_0 \cap K_1$.
Then they define "lift" operators to be some functions $r(\phi) : [L^2(\Gamma)]^2 \rightarrow \Sigma_h$ and $l(\phi) : L^2(\Gamma) \rightarrow \Sigma_h$ such that for all $\tau \in \Sigma_h$,
\begin{gather} \int_\Omega r(\phi) \cdot \tau dV = - \int_\Gamma \phi \cdot \frac{\tau|_{K_0} + \tau|_{K_1}}{2} dS\\ \int_\Omega l(\phi) \cdot \tau dV = - \int_\Gamma \phi (\tau|_{K_0} \cdot \hat{\mathbf{n}}|_{K_0} + \tau|_{K_1} \cdot \hat{\mathbf{n}}|_{K_1}) dS \end{gather} where \begin{gather} \Sigma_h := \{\tau \in [L^2(\Omega)]^2 : \tau|_T \in \Sigma(T)~~~ \forall T \in \{K_0, K_1\}\}\\ \Sigma(T) = [P_p(T)]^2, \end{gather} $P_p(T)$ is the space of polynomial functions of degree at most $p \ge 1$, and $\tau|_K$ indicates the function $\tau$ evaluated at the interface inside of $K$, and $\tau|_T$ indicates the function $\tau$ evaluated at the interface inside of $T$.
As far as I can tell, the function $r(\phi)$ and $l(\phi)$ are just notation to allow a surface integral to be re-written as a volume integral, and I am unsure of what other properties these function have, or even how to evaluate them, but this wasn't detrimental for implementing most methods.
However, now I am trying to make sense of another paper by Peraire et al. which now defines a numerical flux in terms of a "facewise" lift operator (which in the context of this example domain, is exactly the same, i.e. $r^e(\phi) = r(\phi)$ and $l^e(\phi) = l(\phi)$ only because there is exactly one shared interface). In particular, they define (condensed with $C_{11} = 0$ for brevity) \begin{gather} \hat{\mathbf{\sigma}} = \frac{\mathbf{\sigma}^e_h|_{K_0} + \mathbf{\sigma}^e_h|_{K_1}}{2} + \mathbf{C}_{12} (\mathbf{\sigma}^e_h|_{K_0} \cdot \hat{\mathbf{n}}_{K_0} + \mathbf{\sigma}^e_h|_{K_1} \cdot \hat{\mathbf{n}}_{K_1})\\ \mathbf{\sigma}^e_h = \nabla_h u_h + r^e\left( \frac{u_h|_{K_0} + u_h|_{K_1}}{2}\right) + l^e\left( u_h|_{K_0} \hat{\mathbf{n}}|_{K_0} + u_h|_{K_1} \hat{\mathbf{n}}|_{K_1}\right) \end{gather}
To implement this numerically, there is a term which requires me to compute \begin{gather} \int_\Gamma \hat{\sigma} \cdot (v|_{K_0} \hat{\mathbf{n}}|_{K_0} + v|_{K_1} \hat{\mathbf{n}}|_{K_1}) dS \end{gather} where $v$ is a test function defined in $L^2(\Omega)$ restricted to the subsets $K_0$ or $K_1$.
I am at a complete loss as to what it even means to take a surface integral of a lift operator, or even how to do this. So that's my question:
- What are these lift operators $r(\phi)$ and $l(\phi)$, beyond just a convenient way to transform surface integrals into volume integrals?
- How would I go about evaluating the above surface integral of lift operators?