Galerkin method

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Galerkin methods are a family of techniques for turning a continuous operator problem, such as a differential equation written in a weak form, into a finite-dimensional problem by projecting onto a chosen subspace spanned by a finite set of basis functions. They are named after Boris Galerkin.

How it works
- Start with a weak formulation: find u in a function space V such that a(u, v) = f(v) for all v in V, where a is a bilinear form and f is a bounded linear functional.
- Pick a finite-dimensional subspace V_n ⊂ V with basis {e1, e2, ..., en}.
- Seek u_n in V_n that satisfies a(u_n, v_n) = f(v_n) for all v_n in V_n.
- This becomes a linear system A u = f, where u contains the coefficients of u_n in the basis {e_j} and A_ij = a(e_i, e_j). If a is symmetric, A is symmetric.

Galerkin orthogonality and well-posedness
- Let e_n = u − u_n be the error. Then e_n is orthogonal to V_n with respect to a: a(e_n, v_n) = 0 for all v_n in V_n.
- If a is bounded and elliptic, the original problem is well-posed (has a unique solution), and the Galerkin problem is well-posed as well.

Error and best approximation
- Using the energy norm ||w||_a = sqrt(a(w, w)), the error satisfies
||u − u_n||_a ≤ (C / c) inf_{v_n ∈ V_n} ||u − v_n||_a,
where C and c come from the boundedness and ellipticity of a.
- This means u_n is the best approximation to u within V_n in the energy norm (up to a constant). If a is symmetric and positive definite, the energy norm is a true inner product, and u_n is the orthogonal projection of u onto V_n.

Notes
- For nonsymmetric problems, a Petrov–Galerkin variant can be used.
- The Galerkin idea is the foundation of the finite element method, providing a straightforward path from continuous problems to solvable linear systems.