# How to compute $\Delta u$ on the boundary of the biharmonic equation?

Let $u$ be the answer of a PDE.Is there any relationship between $u,\frac{\partial u} {\partial n}$ and $\Delta u$.

I have the values of $u$ and $\frac{\partial u} {\partial n}$ on $\partial \Omega$ but I need the value of $\Delta u$ on the boundary.

I think it's impossible, but is there anyone who know how to do this?

In fact I want to solve biharmonic equation by convert it into two Poisson problems: $$\Delta^2u=f$$ $$u=g_1$$ $$\frac{\partial u} {\partial n}=g_2$$.

Using $\Delta u=w$ leads to $$\Delta u=w,$$$$u=g_1 ~~on~~\partial \Omega$$ $$\Delta w=f ,$$$$w=\Delta u-c(\frac{\partial u} {\partial n}-g_2)~~on~~\partial \Omega$$ so at the first I have to use an initial guess for $\Delta u$on boundary. But by this way the accuracy is low. and sometimes it is dependent to initial guess. Here $c$ is a small constant.

• Why do you need $\Delta u$ on the boundary? Maybe there's a trick that you can use to compute the thing you really need. Jul 4 '14 at 14:24
• If you have the analytical solution to the PDE, you should be able to take the laplacian of it directly. Perhaps you mean to esimate the laplacian on the boundary when $u$ is a numerical solution?
– Paul
Jul 4 '14 at 14:53
• I want to solve biharmonic equation numerically it with Neumann and Dirichlet boundary conditions for $u$. I want to convert this problem into two Poisson equations with Dirichlet boundary condition for both of them. Using $w=\Delta u$
– rosa
Jul 4 '14 at 14:54
• Maybe you could write out the system that you actually want to solve with its full boundary conditions. I think you'll get better responses that way. Jul 4 '14 at 14:58
• This site works best if you comment on your original question if you have difficulty with an answer. You could have asked one of the people answering to explain the specific part of the answer you couldn't implement. Jul 4 '14 at 17:33

If your question is whether you can compute $\Delta u|_{\partial \Omega}$ just from $u|_{\partial \Omega}$ and $\partial_n u|_{\partial \Omega}$ without solving the actual equation, then the answer is no. What you are trying to do is the equivalent of computing the Dirichlet-to-Neumann map for the Laplace equation, which also requires you to compute the solution of the Laplace equation. It is not possible to compute $\Delta u|_{\partial \Omega}$ just from $u|_{\partial \Omega}$ and $\partial_n u|_{\partial \Omega}$ purely locally.

Yes, this can be done but, as you've observed, specifying the boundary conditions for the two Poisson equations can be challenging.

This example from MATLAB PDE Toolbox

Clamped Square Plate

solves the plate equation by converting it to a system of two scalar Poisson's equations. It uses a Robin-type BC as a trick to approximately enforce $u=0$ on the boundary. Specifically, the following two Neumann (Robin) BCs are applied:

$${{\partial u}\over{\partial n}} = g_2$$

and

$${{\partial w}\over{\partial n}} = -ku$$

where $k$ is a large number. If you want $g_1 \ne 0$, it isn't immediately obvious to me how to use this same technique to achieve that.

• thanks, but I could not find the clamped square plate (the webpage). could you please help me in any other way?
– rosa
Jul 4 '14 at 17:17
• That is very strange. Can you access the main MathWorks web page at mathworks.com. If so, you can select the support menu item and navigate down to the PDE Toolbox documentation. Jul 4 '14 at 17:42