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

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

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

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

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

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

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

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

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.