Consider the boundary value problem $-\nabla \cdot (a\nabla u) + cu = f$ in $\Omega$, $u = 0$ on $\partial\Omega$. The weak formulation seeks $u \in H_0^1(\Omega)$ such that $B(u, v) = F(v)$ for all $v \in H_0^1(\Omega)$, where $B(u,v) = \int_\Omega (a\nabla u \cdot \nabla v + cuv)\,dx$ is the bilinear form and $F(v) = \int_\Omega fv\,dx$ is the linear functional. By the Lax-Milgram theorem, if $B$ is continuous and coercive, there exists a unique weak solution.

The Galerkin method restricts the weak formulation to a finite-dimensional subspace $V_h \subset H_0^1(\Omega)$: find $u_h \in V_h$ such that $B(u_h, v_h) = F(v_h)$ for all $v_h \in V_h$. Choosing a basis $\{\phi_1, \ldots, \phi_N\}$ for $V_h$ and writing $u_h = \sum_j c_j \phi_j$ leads to the stiffness system $Kc = f$ where $K_{ij} = B(\phi_j, \phi_i)$ and $f_i = F(\phi_i)$.

On a triangulation $\mathcal{T}_h$ of $\Omega$, the space of continuous piecewise linear (P1) elements is $V_h = \{v \in C(\bar{\Omega}) : v|_T \in \mathcal{P}_1 \text{ for all } T \in \mathcal{T}_h,\, v|_{\partial\Omega} = 0\}$. The hat functions $\phi_i$ form a basis: $\phi_i(x_j) = \delta_{ij}$ at mesh vertices. The stiffness matrix $K$ is sparse (bandwidth $\sim \sqrt{N}$ in 2D) and SPD for coercive $B$. Higher-order elements (P2, P3, ...) use additional degrees of freedom at edge midpoints or interior nodes.