A degradation in accuracy of numerical wave or Helmholtz-type solutions where phase errors accumulate with problem size or frequency, causing global error growth that exceeds local discretization error estimates.

Principle

Occurs because local phase or dispersion errors per element produce cumulative phase shift over many wavelengths or large domains; the error grows with problem size (or wave number) unless discretization is refined beyond standard local error criteria.

Demonstration

When solving the Helmholtz equation at high wave numbers with a fixed-order finite element mesh, the computed solution's phase drifts progressively across the domain, and the L2 error can increase with the wave number despite local element accuracy, illustrating the pollution effect.

Misapplication

Assuming that reducing local truncation error alone (e.g., by smaller elements but same polynomial order) cures the problem can be false; without addressing dispersion and global phase accumulation, errors may remain or even worsen.

Consequence

The pollution effect forces practitioners to use higher-order methods, mesh refinement strategies scaled with wavelength, or dispersion-controlled schemes to regain accuracy; ignoring it yields unreliable wave propagation, scattering, and resonance results.

Reversal

In a pollution-free regime, global errors scale consistently with local discretization error and refining the mesh or increasing polynomial order reduces error predictably; absence of anomalous growth with wave number indicates no pollution.

Boundary

Pertains primarily to wave propagation, Helmholtz, and oscillatory PDEs at moderate-to-high frequencies and to discretizations where phase accuracy matters; it does not refer to unrelated sources of global error such as boundary condition inconsistency or solver tolerance issues.

Semantic Tension

Closely related to numerical dispersion and phase error; the pollution effect emphasizes the global accumulation of those local dispersion errors as problem size or frequency increases, whereas dispersion on its own may be characterized locally.

Synthesis

The pollution effect is the global deterioration of wave-solution accuracy due to cumulative phase/dispersion errors that scale with problem size or wave number; mitigating it requires methods that control dispersion globally, not just locally.