Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be bounded and open
- $u\in L^2(\Lambda,\mathbb R^d)$
It's well-known that there is a weakly differentiable $p\in L_{\text{loc}}^2(\Lambda)$ with $\nabla p\in L^2(\Lambda)$, $$\Delta p=\nabla\cdot u\;,\tag 1$$ i.e. $$\langle\nabla p,\nabla\phi\rangle_{L^2(\Lambda,\:\mathbb R^d)}=\langle u,\nabla\phi\rangle_{L^2(\Lambda,\:\mathbb R^d)}\;\;\;\text{for all }\phi\in C_c^\infty(\Lambda)\;,\tag 2$$ and $$\left.\frac{\partial p}{\partial\nu}\right|_{\partial\Lambda}={\rm n}\cdot u\tag 3\;,$$ where $\nu$ denotes the outer unit normal field of $\partial\Lambda$.
Now, I want to compute $p$ numerically in the case $\Lambda=(0,a)\times(0,b)$ with $a,b>0$.
I'm not searching for a state-of-the-art algorithm; just a simple solver. However, I would be happy about any book/paper where this problem is considered; no matter which method is used.