Finite element methods approximate weak formulations by piecewise polynomial spaces on meshes. They are natural for geometry, boundary conditions, and variational problems.
Weak formulation
Definition. A weak formulation seeks \(u\in V\) such that \(a(u,v)=\ell(v)\) for every test function \(v\in V\). It replaces pointwise derivatives by integral identities and is often the right formulation for nonsmooth solutions.
Galerkin method
Definition. Given a finite-dimensional subspace \(V_h\subset V\), the Galerkin approximation \(u_h\in V_h\) satisfies \(a(u_h,v_h)=\ell(v_h)\) for every \(v_h\in V_h\).
Céa lemma
Theorem. If \(a\) is continuous and coercive, then \(\|u-u_h\|_V\le (M/\alpha)\inf_{v_h\in V_h}\|u-v_h\|_V\). Thus finite element error is controlled by best approximation in the chosen space.
Element basis functions
Proposition. Local basis functions lead to sparse stiffness matrices because most basis functions have disjoint support. Mesh geometry influences both approximation error and conditioning of the resulting linear system.
A posteriori error
Remark. Residual-based a posteriori estimates guide adaptive mesh refinement. They measure where the computed solution fails to satisfy the equation or interface conditions strongly enough.
Weak formulation
For \[ -\nabla\cdot(a\nabla u)+cu=f \quad\text{in }\Omega,\qquad u=0\text{ on }\partial\Omega, \] define \(V=H_0^1(\Omega)\), \[ A(u,v)=\int_\Omega a\nabla u\cdot\nabla v+cu v\,\mathrm{d}x, \qquad F(v)=\int_\Omega fv\,\mathrm{d}x. \] The weak problem is \(A(u,v)=F(v)\) for all \(v\in V\).
Theorem (Lax--Milgram). If \(A\) is bounded and coercive, \[ |A(u,v)|\le M\|u\|_V\|v\|_V,\qquad A(v,v)\ge\alpha\|v\|_V^2, \] then the weak problem has a unique solution and \[ \|u\|_V\le\alpha^{-1}\|F\|_{V^*}. \]
Galerkin approximation and Céa's lemma
Choose a finite-dimensional \(V_h\subset V\) and solve \[ A(u_h,v_h)=F(v_h)\quad(v_h\in V_h). \]
Theorem (Céa). \[ \|u-u_h\|_V \le\frac M\alpha \inf_{v_h\in V_h}\|u-v_h\|_V. \]
Proof. Galerkin orthogonality gives \[ A(u-u_h,v_h)=0\quad(v_h\in V_h). \] For arbitrary \(v_h\), \[ \alpha\|u-u_h\|_V^2 \le A(u-u_h,u-u_h) =A(u-u_h,u-v_h) \le M\|u-u_h\|_V\|u-v_h\|_V. \] Cancel and take the infimum.
Interpolation error
For degree-\(p\) conforming elements on a shape-regular mesh, \[ \inf_{v_h\in V_h}\|u-v_h\|_{H^1} \le Ch^p|u|_{H^{p+1}}. \] Therefore Céa gives the same energy-norm order. An Aubin--Nitsche duality argument can add one power in \(L_2\) when the dual elliptic problem has sufficient regularity: \[ \|u-u_h\|_{L_2}\le Ch^{p+1}|u|_{H^{p+1}}. \]
Assembly and geometry
Local basis support makes the stiffness matrix sparse. Each element integral is pulled back from a reference element by a geometric map; Jacobian factors transform gradients and volume. Quadrature must be accurate enough not to destroy consistency. Shape regularity prevents degenerate elements and keeps inverse inequalities uniform.
Nonconforming and mixed methods
When \(V_h\not\subset V\), Strang's lemma adds a consistency term to best approximation. Mixed methods introduce flux or pressure variables and require an inf--sup condition rather than coercivity. Stabilization can repair an unstable pair, but changes the discrete problem and needs its own consistency analysis.