The Grid Convergence Index
A standardized, conservative discretization-error bar built from a mesh-refinement study, designed to be reported consistently across studies.
A Standard Error Bar
The Grid Convergence Index (GCI) is a widely adopted way to report discretization uncertainty from a mesh-refinement study. It takes the error estimate from Richardson extrapolation and multiplies it by a safety factor, producing a conservative uncertainty band rather than a best-guess error. Its purpose is standardization: different studies computing GCI the same way produce comparable, defensible uncertainty statements.
How It Is Built
- Compute solutions on three systematically refined meshes.
- Determine the observed order of accuracy from the three solutions.
- Apply Richardson extrapolation to estimate the error on the fine mesh.
- Multiply by a safety factor to get the reported uncertainty band.
The Safety Factor
The safety factor accounts for the fact that the extrapolation rests on assumptions that may not hold perfectly. A larger factor is used when only two meshes are available or when the observed order is uncertain; a smaller factor is justified when three meshes confirm the order lies in the asymptotic range. The factor turns a point estimate into a band a reader can treat as a genuine uncertainty.
Checking the Asymptotic Range
GCI is only trustworthy in the asymptotic range, where the leading error term dominates. A useful self-check computes the ratio of the GCI values between the two mesh pairs; a ratio near one, consistent with the observed order, indicates the meshes are in the asymptotic range and the extrapolation is reliable. A ratio far from one warns that the finest mesh is still too coarse and the reported band should not be trusted.
Reported alongside the observed order and the meshes used, the GCI gives a design reviewer a single, conservative number for the discretization contribution to the error budget, computed by a recipe others can check and reproduce.