A provable upper limit on the magnitude of approximation or measurement error under stated assumptions; can be given a priori (before computation) or a posteriori (after computation) and is used to guarantee worst-case performance.

Principle

An error bound quantifies the largest possible deviation between an approximate value and the true value consistent with model assumptions and derivation; a priori bounds depend on theoretical parameters, while a posteriori bounds exploit computed residuals or indicators.

Demonstration

An a priori finite element estimate: for a Poisson problem with sufficient regularity, the H^1-norm error can be bounded by C h^p ||u||_{p+1}, where C is a constant depending on the mesh regularity and p is the polynomial degree; this gives a computable worst-case scaling of error with mesh size.

Misapplication

Using an error bound computed under assumptions that do not hold (e.g., assuming solution smoothness or mesh regularity) and treating the bound as the actual error, or ignoring that constants in the bound may be large and render the bound pessimistic.

Consequence

Valid error bounds provide guaranteed tolerances, stopping criteria, and safety margins for algorithms and measurement protocols; they enable controlled refinement and certification of results within provable limits.

Reversal

Absence of an error bound means there is no provable worst-case guarantee for the approximation; one must then rely on empirical tests, heuristics, or probabilistic error models rather than deterministic guarantees.

Boundary

Applies under the assumptions used in the derivation (regularity, norms, stability); bounds may be pessimistic, asymptotic, or problem-dependent and do not replace empirical validation — they are statements about the worst case, not the typical error.

Semantic Tension

Error bounds compete with typical error estimates: a tight bound is informative, but many bounds are conservative; practitioners must balance reliance on worst-case guarantees with observed average-case behavior.

Synthesis

An error bound gives a defensible ceiling on possible error under explicit assumptions; it is a tool for guaranteeing and controlling computation, best used alongside empirical checks and sensitivity analysis to assess applicability and sharpness.