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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.

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  • $\begingroup$ Can you define your curly and double square brackets in this context? $\endgroup$ – Bill Barth Apr 10 '16 at 16:41
  • $\begingroup$ Hi, I added a note regarding the notation. $\endgroup$ – Martin Vymazal Apr 10 '16 at 17:01
  • $\begingroup$ I don't think you missed the point at all, your idea to represent $\mathbf{r}$ as a linear combination of basis functions and assemble a local linear system is the right one. $\endgroup$ – chris Apr 18 '16 at 8:28
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That seems to be the right way to solve for the lifting operators, see for example Appendix A in "Discontinuous Galerkin methods for the Navier-Stokes equations using solenoidal approximations" by Montlaur et al (there is also an authors copy if you search for the title), it is for the Compact DG Method but as far as I'm aware all of the lifting operators are equivalent up to some small change to modify the behavior slightly (e.g. CDG is a compact modification of the Local Discontinuous Galerkin method)

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