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