Implementing example for solving biharmonic equation in C++

[aneeshs1729]

Dear Developers and Maintainers,
I would like to implement an implementation of the biharmonic equation as a new problem in MFem, and attempt to implement it in base C.

The biharmonic problem is the equation
$\Delta^2u =f(x,y)$.

For my implementation, I will be using clamped boundary conditions.

$u(x,y)=0\in \partial \Omega$, $\nabla u(x,y)\cdot \hat{n}=0\in \partial \Omega$.
The weak form for this PDE is

$\int_{\Omega} \Delta u \Delta v=\int_{\Omega} fv$ .

Please let me know if you have any suggestions regarding this implementation or any specific considerations I should keep in mind.
Looking forward to your feedback!

[jeremylt]

"bakeoff" in this repo is a specific set of test problems for performance comparisons: https://ceed.exascaleproject.org/bps/

This wouldn't be a BP, but you certainly can add a biharmonic equation using MFEM as an example.

[aneeshs1729]

"bakeoff" in this repo is a specific set of test problems for performance comparisons: https://ceed.exascaleproject.org/bps/

This wouldn't be a BP, but you certainly can add a biharmonic equation using MFEM as an example.

So be it then. I’ll add it in MFEM

[jeremylt]

I mean to say that you can add a biharmonic equation example here using MFEM, it just would not be called a "bakeoff" problem.

[rezgarshakeri]

We only support $C^0$ element in libCEED, for biharmonic you need $C^1$. You can introduce new variable and change biharmonic to mixed equation and then use $C^0$ element that we have (see deal.II tutorial)

[jedbrown]

libCEED is algebraic and not limited to $C^0$. One can use $C^1$, but you'll need suitable restrictions and basis tabulations (we don't have a convenience tabulator for Hermite or similar elements).

[rezgarshakeri]

Yeah, adding $C^1$ basis/restriction in libCEED is another option and this paper by Kirby and Mitchell is a good start.

[aneeshs1729]

We only support C 0 element in libCEED, for biharmonic you need C 1 . You can introduce new variable and change biharmonic to mixed equation and then use C 0 element that we have (see deal.II tutorial)

I will use the mixed equation format.

[aneeshs1729]

"bakeoff" in this repo is a specific set of test problems for performance comparisons: https://ceed.exascaleproject.org/bps/
This wouldn't be a BP, but you certainly can add a biharmonic equation using MFEM as an example.

So be it then. I’ll add it in MFEM

I mean in the MFEM examples

[aneeshs1729]

To summarize, here is my proposal and plan for implementation -

My plan is to add my example into the https://github.com/CEED/libCEED/tree/main/examples/mfem directory. My plan is to follow the following steps that is outlined here - using C_0 elements.
Convert the biharmonic equation into 2 systems of ODEs , and then solve them with the technique to solve laplace's equation.

We only support element in libCEED, for biharmonic you need . You can introduce new variable and change biharmonic to mixed equation and then use element that we have (see deal.II tutorial)