Richardson Extrapolation Tool

Finite element verification

Richardson extrapolation for mesh convergence studies

Enter three systematically refined meshes, from coarsest to finest, to estimate the continuum value and the 95% confidence interval using the Grid Convergence Index method.

Use consistent units for mesh edge length and result value.

Input data

Mesh sizes must decrease each row. Result values can be any consistent scalar quantity from your analysis.

Mesh level Element edge length Result value
Coarsest mesh
Medium mesh
Finest mesh

Convergence checks

  • No warnings. The refinement ratios and convergence check are within the usual screening thresholds.

Output data

The extrapolated result is the estimated continuum value based on your finest three meshes.

Estimated exact value
--
Iterations
--
95% lower bound
--
95% upper bound
--
Order of convergence
--
Convergence check
--
Fine-grid GCI
--
Refinement ratios
--

Grid convergence plot

The chart shows the three mesh results and the continuum estimate at zero mesh size.

Valid results will appear here after calculation.

Confidence bounds are shown around the extrapolated exact value.

Interpretation notes

These checks help you decide whether the mesh sequence is good enough for a credible continuum estimate.

  • Keep the mesh sizes in descending order from coarse to fine.
  • Refinement ratios below 1.3 are often too narrow for a stable extrapolation.
  • A convergence check close to 1.0 suggests the solution is approaching the asymptotic range.
  • Oscillatory convergence means the result changes sign between refinements and deserves extra scrutiny.

References: NASA spatial convergence guide and DYNAmore Richardson extrapolation paper.