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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.

Paul
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rosa
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  • 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. – Bill Barth Jul 04 '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 04 '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 04 '14 at 14:54
  • Yes, I need to estimate it. – rosa Jul 04 '14 at 14:56
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    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. – Bill Barth Jul 04 '14 at 14:58
  • You already asked a very similar question four weeks ago (http://scicomp.stackexchange.com/q/12867/1804), and the (accepted) answer explained how to solve this equation. It would be helpful to understand why you are not satisfied with the answers you got then. – Christian Clason Jul 04 '14 at 16:17
  • Yes, I already asked this question, but it was difficult to implement, so I tried to solve it indirectly. – rosa Jul 04 '14 at 17:06
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    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. – Christian Clason Jul 04 '14 at 17:33

2 Answers2

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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.

Wolfgang Bangerth
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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.

Bill Greene
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  • thanks, but I could not find the clamped square plate (the webpage). could you please help me in any other way? – rosa Jul 04 '14 at 17:17
  • That is very strange. Can you access the main MathWorks web page at http://www.mathworks.com/. If so, you can select the support menu item and navigate down to the PDE Toolbox documentation. – Bill Greene Jul 04 '14 at 17:42