Finite Element Method (Mathematical Foundations)

Core Idea

The finite element method (FEM) approximates weak solutions of PDEs by restricting the variational problem to a finite-dimensional subspace V_h ⊂ H¹₀(Ω) built from piecewise polynomial functions on a mesh. The Galerkin approximation u_h ∈ V_h satisfies a(u_h, v_h) = F(v_h) for all v_h ∈ V_h, inheriting the same coercivity and stability as the continuous problem. Cea's lemma guarantees that u_h is quasi-optimal: ||u - u_h|| ≤ (M/α) inf_{v_h ∈ V_h} ||u - v_h||, so the error is controlled by the best approximation error, which vanishes as the mesh is refined.

Explainer

The finite element method is the dominant numerical technique for solving elliptic and parabolic PDEs, and its mathematical theory is a beautiful application of functional analysis. The idea is to replace the infinite-dimensional space H¹₀(Ω) in the weak formulation with a finite-dimensional subspace V_h consisting of piecewise polynomial functions defined on a triangulation (mesh) of Ω. The Galerkin equation a(u_h, v_h) = F(v_h) for all v_h ∈ V_h reduces to a linear system Ku = f, where K is the stiffness matrix.

The mathematical foundation rests on two pillars: the Lax-Milgram theorem (guaranteeing well-posedness of both the continuous and discrete problems) and approximation theory (quantifying how well V_h approximates H¹₀). Cea's lemma bridges them: ||u - u_h||_{H¹} ≤ (M/α) inf_{v_h} ||u - v_h||_{H¹}. The best approximation error is then estimated using interpolation theory: for piecewise polynomials of degree k on a mesh of size h, the interpolation error is ||u - I_h u||_{H¹} ≤ Ch^k |u|_{H^k+1}, where |u|_{H^k+1} measures the (k+1)st derivatives of u.

The Aubin-Nitsche duality argument improves the L² error estimate by one order of h beyond the H¹ estimate. For piecewise linears, this gives ||u - u_h||_{L²} = O(h²) compared to ||u - u_h||_{H¹} = O(h). The argument is indirect: one solves an auxiliary (adjoint) problem and uses its regularity to gain the extra order. This duality technique is a fundamental tool in numerical analysis with applications far beyond the basic Poisson problem.

The finite element framework extends to time-dependent problems (method of lines, space-time FEM), nonlinear PDEs (Newton linearization on each step), and systems. A posteriori error estimation provides computable bounds on the error without knowing the exact solution, enabling adaptive mesh refinement that concentrates computational effort where the error is largest. The mathematical analysis of FEM continues to develop: discontinuous Galerkin methods, mixed methods for constrained problems, isogeometric analysis, and virtual element methods all extend the classical framework while maintaining its rigorous functional-analytic foundation.