The lifting operator $\mathbf{r}(\mathbf{v_h})$ for the '2nd version' of Bassi-Rebay scheme for elliptic problems in $d$ dimensions is defined as $$ \int_{\Omega_h} \mathbf{w}_h \cdot \mathbf{r}(\mathbf{v}_h)\,\mathrm{d}\mathbf{x} = -\int_{E} \{\mathbf{w}_h\}\cdot \mathbf{v}_h \,\mathrm{d}S, $$ where $$ \mathbf{v}_h \in \mathbf{V}_h,\:\mathbf{w}_h \in \mathbf{V}_h\\ V_h = \{v_h \in L^2(\Omega_h) : \bigl. v_h\bigr|_K\in \mathbb{P}_k(K) \:\forall K\in \mathcal{T}_h\}\\ \mathbf{V}_h = \{v_h \in \bigl(L^2(\Omega_h)\bigr)^d : \bigl. v_h\bigr|_K\in \bigl(\mathbb{P}_k(K)\bigr)^d \:\forall K\in \mathcal{T}_h\}, $$ and $E$ is an edge of element belonging to triangulation $\mathcal{T}_h$.
The weak form for Laplace equation discretized with BR2 then contains terms such as $$ \int_{\Omega_h} \bigl(\nabla v_h\bigr) \cdot \mathbf{r}\bigl([\![u_h]\!]\bigr)\,\mathrm{d}\mathbf{x},\quad u_h, v_h \in V_h, $$ for which the definition of the lifting operator can be used to convert the integral over $\Omega_h$ into surface integral over element edge.
There are also face terms such as $$ \int_{E} [\![v_h]\!] \cdot \mathbf{r}\bigl([\![u_h]\!]\bigr)\,\mathrm{d}S. $$ How should these be evaluated?
The only approach I can think of is to use the definition of the lifting operator with all test functions $\mathbf{w}_h$ belonging to one element in order to assemble a local linear system where the expansion coefficients of $\mathbf{r}(\mathbf{v}_h)$ are unknowns (i.e. I would assume that $\mathbf{r}(\mathbf{v}_h)$ can be represented as a linear combination of basis functions). If I do that, then I can work with the lifting operator restricted to element traces. This seems to be a bit strange to me though and I'm afraid I completely missed the point.
Remark - notation: Let $E$ be an internal edge shared by two elements $K^{+}$ and $K^{-}$ and let $\mathbf{n}^{+}$ and $\mathbf{n}^{-}$ be outer normals of $K^{+}$ and $K^{-}$ on this edge. The jump $[\![\cdot]\!]$ and average $\{\cdot\}$ operators are defined as follows:
- $[\![v_h]\!] = v_h^{+}\mathbf{n}^{+} + v_h^{-}\mathbf{n}^{-}$ for $v_h \in V_h$
- $[\![\mathbf{v}_h]\!] = \mathbf{v}_h^{+} \cdot \mathbf{n}^{+} + \mathbf{v}_h^{-} \cdot \mathbf{n}^{-}$ for $\mathbf{v}_h \in \mathbf{V}_h$
- $\{v_h\} = \frac{1}{2}(v_h^{+} + v_h^{-})$, $v_h \in V_h$
- $\{\mathbf{v}_h\} = \frac{1}{2}(\mathbf{v}_h^{+} + \mathbf{v}_h^{-})$, $\mathbf{v}_h \in \mathbf{V}_h$
The superscript ${}^{\pm}$ refers to trace quantities restricted to edge (face in 3D) from $K^{+}$ and $K^{-}$, respectively.
