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This question is a follow-up of this previous one. In "Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems" by Georgoulis et al., an error estimator is provided in equation (4.7).

It contains the term $$||f -b \cdot \nabla \hat u - cu_h - \Pi(f-b \cdot \nabla \hat u-c u_h) ||_T$$

where :

  • $\Pi$ is the local orthogonal $L^2$ projection onto the classical polynomial DG finite element space $S_h= \{v \in L^2: v_{|T} \in \mathbb{P}^{p}(T) \}$

  • $u_h$ is the finite element solution, which I have on each cell $T$

  • $\hat{u}$, called reconstruction, is defined elementwise (see definition 4.3 or the beginning of Section 4) by the relation $$\int_Tb \cdot \nabla \hat{u} v_h = \int_T b \cdot \nabla u_h v_h - \int_{\partial_{-T}} (b \cdot n)[u_h]v_h^{+}$$ for all $v_h \in S_h$ (classical DG fe space)

It's written that the reconstruction space is $\{w_h \in C({\bar{\Omega}}): w_{h|T} \in \mathbb{P}^{p+2}(T)\}$

My big issue is with this reconstruction part!

As far as I understand, I should construct $\hat{u}$ cell-by-cell by using that definition. Of course, as I know $u_h$ and $v_h$, it turns out that a linear system for each cell has to be solved.

However, it seems to me that $\hat{u}$ is in a continuous finite element space, instead of a discontinuous one like $v_h$. So, by expanding the l.h.s I would have

$$(b \cdot \nabla \varphi_j,\phi_i)_T$$

where $\varphi_j$ is a basis function in the continuous finite element space, while $\phi_i$ is a basis function of the discontinuous one $S_h$.

I am a bit confused on how to do this. If I create a different DoFHandler object on the same triangulation, distribute the dofs corresponding to the continuous space, I would not even end up with a square matrix since I don't have the same number of degrees of freedom in these two spaces. So I think I am clearly missing the point. Any help is extremely appreciated.

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  • $\begingroup$ Have you asked the authors whether they really wanted the space to be continuous? It might be a mistake in the paper's writing. $\endgroup$ Aug 23 at 15:36
  • $\begingroup$ @WolfgangBangerth Thanks for your comment. I haven't actually. Why do you think it may be a mistake? $\endgroup$ Aug 23 at 16:06
  • $\begingroup$ Because "cellwise" and "continuous" just doesn't go together. $\endgroup$ Aug 23 at 18:43
  • $\begingroup$ Well, I've just spent a whole day trying to figure out how that could possibly be done. I have also to add that if I try to compute that cellwise matrix (4x4 with bilinear functions) with entries $$(b \cdot \nabla \phi_j,\phi_i)_T$$ with $\{\phi_i\}_i$ DG basis functions, that matrix turns out to be singular. Same thing happens if I try with continuous basis functions. (I tried several fields $b$). In the paper they're using triangles. Is it possible that with quads I have a 4by4 singular matrix, but the 3by3 I'd get with triangles is non singular? @WolfgangBangerth $\endgroup$ Aug 23 at 20:05
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    $\begingroup$ I mean it's singular because I computed its determinant and it's about $10^{-40}$, which means that I would obtain indeed 0 for a constant $\phi_j$. I'm going to ask them and will update this question. Thanks Prof. @WolfgangBangerth $\endgroup$ Aug 24 at 7:01

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