Functional Analysis in PDEs

Ansys, Abaqus, OpenFOAM, FEniCS - every industrial PDE solver runs on the finite element method. FEM in turn is one theorem deep: weak formulation in a Sobolev space plus Lax-Milgram, and the rest is bookkeeping.

• **Physical simulation**: FEM solves elasticity, heat conduction, fluid dynamics; convergence O(h^k) as h -> 0 is mathematically rigorous
• **Physics-Informed Neural Networks**: PINNs minimize the variational residual integral |(nabla u . nabla v - f v)|^2; this is the variational FEM form with the network as the Galerkin space V_h
• **Finite element graph (FEG)**: GNNs with FEM basis functions as aggregation weights - direct fusion of deep learning and numerical FEM

Предварительные знания

• Distribution Theory

Weak Formulation

**Weak formulation**: instead of solving -Delta u = f classically, seek u in H^1_0(Omega) such that integral nabla u . nabla v dx = integral f v dx for all v in H^1_0(Omega). This is the weak (variational) form. Multiplying by a test function v and integrating by parts transfers derivatives from u onto v.

**Bilinear form**: a(u,v) = integral nabla u . nabla v dx. Linear functional: L(v) = integral f v dx. Problem: find u in H^1_0 such that a(u,v) = L(v) for all v in H^1_0. This is an abstract problem on a Hilbert space, to which the Lax-Milgram lemma applies.

Lax-Milgram Lemma

**Lax-Milgram lemma**: let H be a Hilbert space, a: H x H -> R a continuous coercive bilinear form (|a(u,v)| <= M||u||||v||, a(u,u) >= alpha||u||^2), and L in H* a bounded linear functional. Then there exists a unique u in H such that a(u,v) = L(v) for all v in H.

**Coercivity for the Poisson equation**: a(u,u) = integral |nabla u|^2 = |u|^2_{H^1} >= (1/C_P^2)||u||^2_{H^1} by the Poincare inequality. This gives alpha = 1/C_P^2 > 0, making the Lax-Milgram lemma applicable.

Lax-Milgram generalizes the Riesz representation theorem: in the symmetric case a(u,v) = a(v,u) the lemma follows directly from Riesz. For non-symmetric forms (convection-diffusion equations) the full Lax-Milgram is needed.

Galerkin Method and FEM

**Galerkin method**: instead of the infinite-dimensional H^1_0, seek an approximate solution in a finite-dimensional subspace V_h of H^1_0. Problem: find u_h in V_h such that a(u_h, v_h) = L(v_h) for all v_h in V_h. This becomes a linear system K*u = F, where K_ij = a(phi_j, phi_i) and F_i = L(phi_i).

**Ceea estimate (Galerkin quasi-optimality)**: ||u - u_h||_{H^1} <= (M/alpha) * inf_{v_h in V_h} ||u - v_h||_{H^1}. FEM (finite element method) takes V_h to be piecewise polynomial on a triangulation of Omega. As h -> 0: ||u - u_h||_{H^1} = O(h^k) for degree-k elements.

Key Ideas

• **Weak formulation**: multiply -Delta u = f by v, integrate by parts -> a(u,v) = L(v) for all v in H^1_0; reduces regularity requirements
• **Lax-Milgram**: continuous + coercive a + bounded L => unique u; Poincare inequality guarantees coercivity for the Poisson equation
• **FEM (Galerkin)**: seek u_h in V_h of H^1_0; stiffness matrix K + load vector F -> linear system; P1 elements: O(h) in H^1, O(h^2) in L^2

Related Topics

FEM unifies all of functional analysis:

• Sobolev Spaces — H^1_0 is the admissible function space; embedding theorems guarantee regularity of the finite element solution
• Functional Analysis in ML — RKHS and Neural Tangent Kernel - the functional space of neural networks; the analogue of the Galerkin space V_h

Вопросы для размышления

• PINNs use a variational formulation. What is the role of the functional space in which the loss is minimized? What does the Lax-Milgram lemma guarantee for PINNs?
• The Galerkin method reduces an infinite-dimensional problem to a finite-dimensional linear system. How does the choice of basis functions V_h affect the condition number of the stiffness matrix K?
• For convection-diffusion equations (with a convective term b . nabla u), coercivity fails at large Peclet numbers. How do stabilized FEM methods (SUPG) restore applicability of the Lax-Milgram lemma?

Связанные уроки

• calc-17-multivariable