##  [Collocation Method](/collocation-method-0) 

 Definition

A numerical technique that approximates the solution of differential or integral equations by enforcing the governing equations exactly at a selected finite set of collocation points and solving for coefficients in a chosen finite basis.

 

 

 

 

 

 





## Principle

Principle

Represent the unknown as a finite linear combination of basis functions and require the residual to vanish at chosen collocation points, converting a continuous problem into a finite algebraic system for the coefficients.

 

 

 

 

 





## Demonstration

Demonstration

Approximate a boundary-value ODE by expressing the solution in a polynomial basis, choose Gauss–Lobatto collocation points, enforce the differential equation and boundary conditions at those points, and solve the resulting linear system for polynomial coefficients.

 

 

 

 

## Misapplication

Misapplication

Picking collocation points poorly (clustering excessively or ignoring endpoint behavior) or using an ill-suited basis can produce large interpolation errors, Runge phenomena, or an ill-conditioned algebraic system.

 

 

 

 

 





## Consequence

Consequence

When implemented with appropriate basis and point selection, collocation yields high-order convergence and sparse structured systems that are efficient for many boundary-value and integral problems.

 

 

 

 

## Reversal

Reversal

Contrasts with Galerkin projection: collocation enforces pointwise residual cancellation at discrete locations, while Galerkin enforces orthogonality of the residual against a test space.

 

 

 

 

 





## Boundary

Boundary

Effective for smooth solutions and problems amenable to global or piecewise basis representations; less robust for problems with strong discontinuities, non-smooth coefficients, or where pointwise enforcement breaks weak formulations.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between pointwise enforcement (collocation) and weighted-average enforcement (Galerkin): collocation is simpler and computationally cheaper per degree of freedom but can be less stable or less respectful of weak formulations.

 

 

 

 

 





## Synthesis

Synthesis

The collocation method reduces continuous operator equations to finite algebraic systems by choosing basis functions and enforcing the governing equations at selected points, trading pointwise accuracy for a discrete solvable system.