Biharmonic equation formulation

variational formulation
1

Hello,

I am currently trying to solve a biharmonic-like equation in Fenics.

I am following the provided tutorial (https://fenicsproject.org/docs/dolfin/2019.1.0/python/demos/biharmonic/demo_biharmonic.py.html) where it presents the variational formulation. It uses a C0 interior penalty method, and after some research another question in the forum (https://fenicsproject.org/qa/6281/weak-formulation-of-biharmonic-equation/) pointed me to the article by Brenner and Sung “C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains”.

However, in that article the penalty terms are included with a positive sign:
equation
whereas in the tutorial the second and third penalty terms are negative (they are equivalent to the term b_h in the mentioned article).

I would like to know if this is a mistake, or if the sign changes due to some internal behaviour of Fenics (maybe the way it provides the interior boundary values or some other sign convention).

Thank you very much.

1 Like
2

I believe this is just a sign convention. If you look at the text above (3.3) in Brenner and Sung, my understanding of

n_e is pointing from T_- to T_+

is that n_e is n_- in the notation of the FEniCS biharmonic demo. Thus,

\begin{array}{rl} [\![w]\!]_{\text{B--S}} &= w_+\cdot n_e - w_-\cdot n_e\\ ~&= w_+\cdot n_- - w_-\cdot n_- \\ ~&= -(w_+\cdot (-n_-) + w_-\cdot n_-)\\ ~&= -(w_+\cdot n_+ + w_-\cdot n_-)\\ ~&= -[\![w]\!]_{\text{FEniCS}}\text{ ,} \end{array}

where [\![\cdot]\!]_{\text{B--S}} and [\![\cdot]\!]_{\text{FEniCS}} are the definitions of “[\![\cdot]\!]” from Brenner and Sung and the FEniCS demo respectively.

2 Likes
3

Thank you kamensky,

You are completely right. I was asuming that n_e was n_+ in FEniCS notation. I was also confused by the fact that in Brenner and Sung they mention that the value of the operator does not depend on the choice of n_e.

Thanks again.