### Curvature boundary condition for 2nd order PDE

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Hello everyone,

I am trying to implement a curvature-based boundary condition for a 2nd order PDE problem. Without going deeply into details, consider we have a system like this (not exactly like I actually have):

\[au+bu'+cu''=0 \ \ on \ \ \Omega,\\-bu-cu'=\kappa \ \ on \ \ \partial \Omega_{\kappa} ,\\ \kappa =u''\]

Essentially, I have a 2D problem instead of 1D. \(\kappa\) here represents the surface curvature, which arises from a surface tension boundary condition.

As you can see, the usual imposition of the boundary condition will lead to the weak form order increase by one (since we substitute 0 and 1st order boundary terms with 2nd order curvature term).

\[

0 = \int (v,au) - (v', bu + c u') + \int_{\partial \Omega} (v, bu + cu') \rightarrow \int (v,au) - (v', bu + c u') - \int_{\partial \Omega} (v, u'')

\]

My question is: is it possible to do this in FEM context or should I find a way to somehow remove the 2nd order curvature term?

Thank you in advance!

I am trying to implement a curvature-based boundary condition for a 2nd order PDE problem. Without going deeply into details, consider we have a system like this (not exactly like I actually have):

\[au+bu'+cu''=0 \ \ on \ \ \Omega,\\-bu-cu'=\kappa \ \ on \ \ \partial \Omega_{\kappa} ,\\ \kappa =u''\]

Essentially, I have a 2D problem instead of 1D. \(\kappa\) here represents the surface curvature, which arises from a surface tension boundary condition.

As you can see, the usual imposition of the boundary condition will lead to the weak form order increase by one (since we substitute 0 and 1st order boundary terms with 2nd order curvature term).

\[

0 = \int (v,au) - (v', bu + c u') + \int_{\partial \Omega} (v, bu + cu') \rightarrow \int (v,au) - (v', bu + c u') - \int_{\partial \Omega} (v, u'')

\]

My question is: is it possible to do this in FEM context or should I find a way to somehow remove the 2nd order curvature term?

Thank you in advance!

Community: FEniCS Project

### 1 Answer

0

Hi there,

reviewing the answer of this post may give you some more ideas about the curved boundaries, at least something to start looking

https://fenicsproject.org/qa/12499/suggestions-increasing-assemble-different-approach-integration

good luck!!

reviewing the answer of this post may give you some more ideas about the curved boundaries, at least something to start looking

https://fenicsproject.org/qa/12499/suggestions-increasing-assemble-different-approach-integration

good luck!!

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thank you for the link, but I'm not quite sure I understand how are curved boundaries related to my question.