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.