I seek to solve an eigenproblem like:
$\Delta U+\lambda U=0$
$U|_{\partial \Gamma}=0$
I want to "try" solving it for initial guess $\lambda_{0}$. Suppose that equation for this guess holds within $\varepsilon$ and produces a function $U_{0}$. I can try increasing or decreasing $\lambda$ and see if $\varepsilon$ lowers or increaces, thus building an iterative process of finging $\lambda$. Is there a sensible way to know how much should I update it? For example, if it was a Schrodinger equation a good guess would be to calculate $E_{0}=\left\langle U_{0}|\hat{H}|U_{0}\right\rangle$ for whatever function I get and update $\lambda_{1}=E_{0}$ accordingly.
