While looking at step6 of deal.ii tutorials, I decided to try to understand how the constraints coming from hanging nodes are imposed. So I started by watching video lecture 16 by prof. Bangerth

As far as I understood: some of the basis functions $\varphi_i$ are now discontinuous, and that's of course a problem as we want to compute $L^2$ products of gradients on local cells. Also, our space is no more a subspace of $H^1$. In practice, what they do is to ensure that every linear combination $\sum_i V_i \varphi(x)$ of such basis functions is indeed continuous, by playing with the cofficients and imposing continuity at the hanging nodes.

What I can't understand is the sentence at 10:10, when basically he says:

I need that the value of the function $v_h$ at vertex 1 is one half the value at vertex 0 plus one half the value at vertex 2

I know this follows by imposing continuity, but I can't see the computation behind this. I can't see how the continuity is imposed, mathematically.

  • $\begingroup$ Maybe the more extensive discussion in Section 4.3 of this paper helps clarify things: math.colostate.edu/~bangerth/publications/2006-hp.pdf $\endgroup$ Jun 10 at 16:19
  • $\begingroup$ Also section 4.4. $\endgroup$ Jun 10 at 16:20
  • $\begingroup$ Thanks @WolfgangBangerth. So, in my case the continuity requirement is precisely $$V_1 \phi_1(x) = V_0 \phi_0(x) + V_2 \phi_2(x)$$ for $x$ in the line from 0 to 2 (I'm looking at the numbering in the video-lecture). What I really can't see is where the above equation is coming from. I mean, why does that correspond to continuity of a function in $V_h$? $\endgroup$ Jun 10 at 16:44
  • $\begingroup$ The left hand side $V_1\phi_1(x)$ is the value at a point $x$ on the edge as seen from the two small cells, specifically at the end midpoint. The right hand side is the value at that edge midpoint as seen from the big cells. Requiring continuity means that the two need to be the same. $\endgroup$ Jun 10 at 18:13
  • $\begingroup$ @WolfgangBangerth Thanks for your comment. What do you mean with: "specifically at the end midpoint"? If $V_1 \phi_1(x)$ is $v_h(x)$ restricted to the line 0-1-2 as seen from the two small cells, why don't I have also $\phi_0(x), \phi_2(x)$? It seems that their support is not on the two small cells, but only on the big one on the bottom left. Is that correct? $\endgroup$ Jun 10 at 23:28

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