##  [Galerkin Method](/galerkin-method-2) 

 Definition

A projection-based approximation technique for differential or integral equations that seeks approximate solutions in a finite-dimensional subspace by imposing that the residual be orthogonal to that subspace (or a test subspace).

 

 

 

 

 

 





## Principle

Principle

Replace an infinite-dimensional variational problem by a finite-dimensional one: choose trial and test spaces, project the residual to zero on the test space, and obtain solvable algebraic systems whose solutions approximate the true solution under consistency and stability conditions.

 

 

 

 

 





## Demonstration

Demonstration

To approximate the Poisson problem −Δu=f on a bounded domain, pick a finite basis of H^1_0(Ω) (e.g. piecewise linear hat functions) and require ∫Ω ∇u_h·∇v_h = ∫Ω f v_h for every basis test function v_h; assembling yields a sparse linear system for the coefficients of u_h.

 

 

 

 

## Misapplication

Misapplication

Using non-conforming trial/test spaces, ignoring compatibility (inf-sup) conditions for mixed problems, or failing to control interpolation/stability leads to non-convergent schemes or spurious modes.

 

 

 

 

 





## Consequence

Consequence

Under appropriate approximation properties and stability (Galerkin orthogonality, Céa's lemma), the method yields convergent approximations with quantifiable error bounds and a practical route to numerical solution via matrix assembly and linear solvers.

 

 

 

 

## Reversal

Reversal

Choosing test spaces different from trial spaces yields Petrov–Galerkin schemes with different stability characteristics; collocation or least-squares methods abandon orthogonality in favor of pointwise or least-squares residual control.

 

 

 

 

 





## Boundary

Boundary

Applies to variational formulations on Hilbert or Banach spaces where suitable finite-dimensional subspaces exist; excludes schemes without projection structure (pure collocation without variational underpinning) or problems lacking a well-posed weak form.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often contrasted with collocation, least-squares or spectral methods: Galerkin enforces weak orthogonality and inherits functional-analytic stability, while other approaches prioritize pointwise matching or global basis properties.

 

 

 

 

 





## Synthesis

Synthesis

The Galerkin method projects an infinite-dimensional variational problem onto finite-dimensional subspaces, producing algebraic systems whose solutions approximate the true solution when consistency and stability (approximation and inf-sup) requirements are met.