2.1 Poisson Problem

We want to calculate a numerical solution $U$, which approximates the true solution $u$ of the Poisson (boundary value) problem $(P)$ in the solution domain $\Omega \subset \mathbb{R}^3$ with closed boundary $\Gamma$. Dirichlet boundary conditions apply on $\Gamma_D \subset \Gamma$ and Neumann boundary conditions apply on $\Gamma_N:=\Gamma \backslash \Gamma_D$.

The Poisson problem $(P)$ is defined as follows: Given $f \in L^2(\Omega)$, $u_D \in H^1(\Omega)$, and $g \in L^2(\Gamma_N)$, we are searching for the solution $u \in H^1(\Omega)$, which satisfies the Poisson equation

$$-\Delta u = f \quad \mathrm{in } \Omega$$ (2.1)

with Dirichlet boundary conditions

$$u = u_D \quad \mathrm{on } \Gamma_D$$ (2.2)

and Neumann boundary conditions

$$\frac{\partial u}{\partial n}=g \quad \mathrm{on } \Gamma_N \quad.$$ (2.3)