Galerkin Methods — Mathematics Knowledge System

Overview

Galerkin methods approximate the weak solution of a PDE by projecting the problem onto a finite-dimensional subspace V_h of the Sobolev space V. The Galerkin approximation u_h in V_h satisfies a(u_h, v_h) = L(v_h) for all v_h in V_h — the same weak formulation but restricted to a finite-dimensional space. The Cea lemma gives the best-approximation error estimate: ||u - u_h|| <= (M/alpha) inf_{v_h in V_h} ||u - v_h||. Finite element methods are the most important special case.

Intuition

Galerkin methods convert an infinite-dimensional problem (find u in V) into a finite-dimensional one (find u_h in V_h) by only testing against functions in V_h. The key insight is that even though we test with fewer functions, the error ||u - u_h|| is controlled by how well V_h approximates u. The Galerkin approximation is the best approximation from V_h in the energy norm (up to a constant), making it highly efficient when V_h is chosen to match the local structure of the solution.

Formal Definition

Definition

Given the weak problem: find u in V with a(u,v) = L(v) for all v in V, the Galerkin approximation for a subspace V_h subset V is: find u_h in V_h with a(u_h, v_h) = L(v_h) for all v_h in V_h. Since V_h is finite-dimensional with dim V_h = N, this is equivalent to an N x N linear system. If V_h = span{phi_1,...,phi_N}, then u_h = sum_j u_j phi_j where the u_j satisfy the stiffness matrix equation: A u = b, where A_{ij} = a(phi_j, phi_i) and b_i = L(phi_i).

a(u_h, v_h) = L(v_h) \quad \forall v_h \in V_h \subset V

Galerkin formulation

\|u - u_h\|_V \le \frac{M}{\alpha}\inf_{v_h \in V_h}\|u - v_h\|_V

Cea's lemma

\sum_j u_j\, a(\phi_j, \phi_i) = L(\phi_i),\quad i = 1,\ldots,N

Stiffness matrix system

Notation

Notation	Meaning
$V_h$	Finite-dimensional approximation subspace
$u_h$	Galerkin approximation to u
$A_{ij} = a(\phi_j,\phi_i)$	Stiffness matrix entries
$h$	Mesh parameter (in finite element context)

Theorems

Theorem 12.1: Theorem 12.1 (Cea's Lemma)

If a is continuous (constant M) and coercive (constant alpha) on V, and V_h is any finite-dimensional subspace of V, then the Galerkin approximation u_h satisfies ||u - u_h||_V <= (M/alpha) inf_{v_h in V_h} ||u - v_h||_V. If a is symmetric (hence an inner product up to rescaling), then M/alpha = 1 and u_h is the best approximation from V_h in the energy norm a(·,·)^{1/2}.

Theorem 12.2: Theorem 12.2 (Galerkin Orthogonality)

The Galerkin error e_h = u - u_h is orthogonal to V_h in the bilinear form: a(u - u_h, v_h) = 0 for all v_h in V_h. This is the key identity behind Cea's lemma.

Theorem 12.3: Theorem 12.3 (Convergence)

If V_h -> V as h -> 0 in the sense that inf_{v_h in V_h} ||u - v_h|| -> 0 for all u in V, then the Galerkin approximations u_h converge to u: ||u - u_h|| -> 0 as h -> 0.

Worked Examples

1
Weak formulation: integral_0^1 u'v' = integral_0^1 fv for all v in H^1_0[0,1]. Here a(u,v) = integral u'v'.
2
Basis: phi_1 = sin(pi x), phi_2 = sin(2 pi x). Stiffness matrix A_{ij} = integral phi_j' phi_i'.
3
Compute: A_{11} = integral_0^1 (pi cos(pi x))^2 dx = pi^2/2. A_{22} = integral_0^1 (2pi cos(2pi x))^2 dx = 2pi^2. A_{12} = A_{21} = 0 (orthogonality of different sine modes).
$A = \begin{pmatrix} \pi^2/2 & 0 \\ 0 & 2\pi^2 \end{pmatrix}$
4
The system is diagonal: u_1 = b_1/(pi^2/2) = (2/pi^2)integral_0^1 f sin(pi x) dx, and u_2 = b_2/(2pi^2) = (1/(2pi^2))integral_0^1 f sin(2pi x) dx.

✓ Answer

For V_h = span{sin(pi x), sin(2pi x)}, the Galerkin system decouples by orthogonality of sine functions, giving u_h = u_1 sin(pi x) + u_2 sin(2pi x) with coefficients determined by projecting f onto these modes.

Practice Problems

Mediumproof writing

Prove Cea's lemma: ||u - u_h||_V <= (M/alpha) inf_{v_h in V_h} ||u - v_h||_V.

Mediumfree response

Explain why the stiffness matrix A_{ij} = a(phi_j, phi_i) is sparse for finite element bases with local support.

Quiz

Summary

Galerkin methods find u_h in V_h subset V satisfying a(u_h,v_h) = L(v_h) for all v_h in V_h.
Galerkin orthogonality: a(u - u_h, v_h) = 0; the error is a-orthogonal to V_h.
Cea's lemma: ||u - u_h|| <= (M/alpha) inf_{v_h in V_h}||u - v_h||.
Finite element methods are Galerkin methods with piecewise polynomial basis functions on a mesh.
The stiffness matrix A_{ij} = a(phi_j, phi_i) is sparse for FEM bases with local support.

References

BookBrenner, S. & Scott, L.R. — The Mathematical Theory of Finite Element Methods (3rd ed.), Springer, 2008
BookErn, A. & Guermond, J.-L. — Theory and Practice of Finite Elements, Springer, 2004
Websiteen.wikipedia.org