Poisson equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}): \Omega \to \mathbb{R}~ \text{such that} \\ &\mathbb{BVP}_u\begin{cases} \nabla^2 u = f & \forall\textbf{x}\in\Omega \\ u=u_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}} \\ \textbf{n}\cdot\nabla{u} = u_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_u \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain} \\ u_{\text{D}}(\textbf{x})~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition}\\ u_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition}\\ f(\textbf{x}) & \text{forcing}\\ \end{cases} \end{align*} \end{split}\]

Weak form

\[\begin{split} \begin{align*} &\text{Find}~u\in V~\text{such that} \\ &F(u, v)=-\int_\Omega\text{d}\Omega~\nabla v\cdot\nabla u + vf + \int_{\partial\Omega_{\text{N}}}\text{d}\Gamma~vu_{\text{N}}=0\quad\forall v\in V~. \end{align*} \end{split}\]

Linear algebra

\[\begin{split} \begin{align*} &v=\xi_i~,~u=\sum_jU_j\xi_j\implies A_{ij}U_j=b_i \iff \mathsf{A}\cdot\textbf{U}=\textbf{b} \\ &\text{where} \\ &A_{ij} = \dots \\ &b_i = \dots \end{align*} \end{split}\]