How do you build a polyharmonic discrete system?

Question

Polyharmonic equations, to my understanding, are defined as:

$$\Delta ^k u = 0$$

i.e. one repeatedly applies the laplace operator to the function a certain number of times and the result must be 0.

The system can be discretized quite easily using finite differences or in the case I care about, triangle meshes, using the discrete Laplace-Beltrami operator.

However, this system is incomplete, we must also have boundary conditions, else there is a trivial solution yielded by $u = 0$.

So, let's say I am given a mesh $M$. I can construct a matrix $L$ through the use of the laplace beltrami oerator. But this is only possible at interior (not boundary) vertices.

Let's say I know which vertices in my mesh are fixed, what do I do to construct an appropriate linear system to solve the polyharmonic problem in the discrete setting?

Answer 1

This is, ultimately, a modeling question: The boundary conditions you choose are the ones that match the physical situation you are trying to describe. For example, the biharmonic equation ($\Delta^2$) describes the vertical deflection of thin plates, and among the boundary conditions you can choose are the "simply supported" case (the edge of the plate just rests on a support) and the "clamped" case (the edge of the plate is clamped). In one case, you prescribe $u$ and $\Delta u$ at the boundary, in the other you prescribe $u$ and $\partial u/\partial n$.

Which one of these is correct depends of course on what you want to do.