Error, Stability, and Convergence

Analyzes numerical error, stability, and convergence through refinement and reference comparisons.

Conceptual context

A simulation may produce smooth numbers, attractive graphs, and repeatable outputs while still being wrong. Reliability requires separating three questions: how far the result is from a reference, whether small perturbations grow during computation, and whether the solution improves under refinement. $E_a$ and $E_r$ measure discrepancy, $G$ diagnoses stability, and $p$ describes how $E$ decreases as $h$ is reduced. Reliability emerges when these pieces agree with one another and with the physical model.

🟢 Essential level

Error measures the distance between $u_h$ and $u_r$. It may retain physical units through $E_a$ or be normalized through $E_r$.

Stability asks what happens to a small perturbation: if it grows from step to step, numerical noise may dominate the calculation. Convergence asks whether the result improves when $h$ decreases. A reliable solution should show decreasing error, stable behavior, and a reproducible refinement pattern. One value is never enough. The same norm, reference, model, and solver tolerance must be retained. If error stops falling, the solver, roundoff, or physical model may dominate instead of the mesh. Each refinement should therefore be treated as a controlled experiment in which only resolution changes and the expected trend is tested. Comparing several resolutions reveals whether improvement is systematic rather than accidental.

Before accepting any error figure, it is worth asking which reference it is measured against, with which norm, and under which solver settings, because the same number can represent an excellent or an insufficient result depending on that context.

🔵 Formal level

Basic formulas

Absolute error is defined by:

Absolute error

The amplitude of the asymptotic law is set by $C$, which depends on the method, regularity, and chosen norm.

Normalized comparison uses:

Relative error

In the asymptotic regime, discretization error follows a power law:

Order-p convergence law

The exponent $p$ is estimated from $E_1$, $E_2$, $h_1$, and $h_2$. On log-log axes, the straight-segment slope is the observed order.

Real student questions

Does small error prove the model is correct? No. It proves closeness only to the chosen reference.

Does stable mean exact? No. It means perturbations do not grow uncontrollably.

Why are more than two meshes needed? Two points always define a slope, even outside the asymptotic regime.

What does a plateau mean? Another error source dominates or the reference cannot resolve further improvement.

Is higher order always better? It reduces error faster asymptotically but may cost more or have a larger error constant.

Final synthesis

A reliable solution is not identified by one number. It needs well-defined error, perturbation stability, convergence under refinement, and a budget separating discretization, solver, roundoff, and model contributions. Reproducible evidence matters more than visual smoothness.