Variational formulation in the DG Poisson demo

Question

I'm trying to understand the undocumented poisson demo with interior penalty. The beginning of the quadratic form a is

a = dot(grad(v), grad(u))*dx \
   - dot(avg(grad(v)), jump(u, n))*dS \
   - dot(jump(v, n), avg(grad(u)))*dS \

which, as far as I can see, can be rewritten to a sum
$$
\sum_{K\in \mathcal{T}_h} (\int_K \nabla\,u\cdot\nabla v dV -\int_{\partial K} v \lbrace\nabla u\rbrace\,\cdot n dS - \int_{\partial K} \, u\,\lbrace \nabla v\rbrace\cdot n dS)
$$
What is the meaning of the last term with the trial function $u$ with no derivative? Any reference for a better understanding? Thanks..
Peter

Answer 1

which, as far as I can see, can be rewritten to a sum

Actually it can't be rewritten this way as neither $u$ and $v$ are well-defined on any $\partial K$.

The point is that the term dot(avg(grad(v)), jump(u, n))*dS is consistent with a solution of the original problem so it can be added. The significant advantage is that it makes your form symmetric. On the other hand I've seen it also being added with opposite sign which introduced another advantage - I can't remember which-like.

Comments

Flipping the sign can can have stability advantages with the two non-symmetric terms balancing each other in terms of coercivity. A drawback is the loss of adjoint consistency, which leads to sub-optimal convergence in the $L^2$ norm, see http://dx.doi.org/10.1137/S0036142901384162

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@JanBlechta "Can be rewritten", I thought "u" and "v" on the interior side of each cell K; however, thanks for the commentary. I see it is symmetric and consistent.

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Surely, you're correct. My mistake.

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Hello,

Can someone please edit or remove the comment that contains the link to this siam article (see http://dx.doi.org/10.1137/S0036142901384162

In the original comment, the ) and . at the end is contained in the link and causes a crossref doi error to be reported. Please remove the ) and . from the comment or delete it, so no further errors are generated.

thanks, SIAM

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Link has been fixed.

Answer 2

The demo uses the standard interior penalty method. See http://dx.doi.org/10.1137/S0036142901384162 (also at http://umn.edu/~arnold/papers/dgerr.pdf ) and the references therein.