# Numerical iterative methods for Poisson equation

Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$, but we don't know what $f$ actually is), and in addition we may know one of the following,

1. discrete data points $({\bf x}_1,\varphi_1({\bf x})),...,({\bf x}_k,\varphi_k({\bf x}))$
2. boundary values $\varphi({\bf x})$ for any ${\bf x}\in\partial \Omega$

The task is to evaluate $\varphi({\bf x})$ given an ${\bf x}\in \Omega$ using an iterative algorithms. Please help suggest potential iterative algorithms, any references to books or publications is helpful. Thanks!

• this question might be well suited for Computational Science SE: scicomp.stackexchange.com Mar 14 '18 at 5:20