Grid Convergence Index (GCI) Calculator

How to Perform a Mesh Convergence Study Using the Grid Convergence Index (GCI)

A CFD result is only as trustworthy as the mesh it was computed on. Refine the mesh too little and your solution carries hidden discretization error. Refine too much and you waste computational resources on accuracy you already had. The Grid Convergence Index gives you a systematic, standardized way to quantify exactly how much mesh-related uncertainty your result carries — and to estimate the grid-independent solution without running an infinitely fine mesh.

This guide explains how to set up a proper mesh convergence study, how to interpret the results from the calculator above, and what to do when things don’t go as expected.

Precision vs. Accuracy — What GCI Actually Tells You

Before diving into the details, it is worth being honest about what the GCI can and cannot do. A common misconception is that a low GCI means your simulation is accurate. It does not. The GCI is a measure of precision, not accuracy.

Think of it this way: mesh refinement improves how certain you are of your result, but it does not guarantee that the result is correct. If your boundary conditions are wrong, your turbulence model is a poor fit for the flow, or your domain is too small, you will converge toward the wrong answer with high confidence. The GCI will happily report a tight error band around a value that has nothing to do with reality.

This distinction matters in practice. A GCI of 0.5 % tells you that further mesh refinement would not change your result by more than about half a percent. It does not tell you that your result matches the physical world to within half a percent — that depends on the combined effect of your models, boundary conditions, numerical schemes, and all the other choices you made when setting up the simulation.

The mesh convergence study therefore sits within a larger verification and validation framework. GCI handles the verification part (is the numerical solution converging as expected?), while validation requires comparison against experimental data or higher-fidelity simulations. Both are necessary to build confidence in your results.

That said, among all the uncertainties in a typical CFD simulation, the mesh-induced uncertainty is one of the few that you can actually quantify rigorously. This makes the GCI one of the most valuable tools in your arsenal, even with its limitations.

Setting Up Your Grid Study

Choosing Three Mesh Levels

You need at least three systematically refined meshes. The key word is systematic — you should not simply remesh with different settings each time. Instead, change a single global parameter (such as the base size in Simcenter STAR-CCM+) by a constant factor between levels.

A refinement ratio of √2 ≈ 1.41 is a practical choice for 3D simulations. This roughly doubles the cell count in 3D (since N scales as h⁻³), which is a meaningful change without being computationally prohibitive. A ratio of 2.0 gives a more decisive test but quadruples the cell count in 3D, which may be too expensive for large industrial models.

The minimum recommended refinement ratio is 1.3. Below this, the difference in cell size is too small to reliably separate discretization error from other error sources like iterative convergence or round-off. A refinement ratio very close to 1.0 is a common pitfall — it can produce misleadingly low GCI values simply because the solutions barely differ, not because the mesh is adequate.

What to Keep Constant

When refining the mesh, certain settings must remain identical across all three levels to ensure you are testing spatial discretization only:

Prism layer settings. The number of prism layers, stretching ratio, and total thickness (or target y⁺) should not change between meshes. Changing prism layers would simultaneously alter the near-wall resolution and the volume mesh, making it impossible to isolate the discretization error. This means you should target y⁺ ≈ 1 already on the coarsest grid — not the finest.

Physics and solver settings. Turbulence model, boundary conditions, time step (for transient cases), convergence criteria, and all other simulation parameters must be identical.

Domain and geometry. The computational domain, CAD geometry, and all geometric features should be the same across meshes.

The only thing that changes is the characteristic cell size in the volume mesh.

Choosing Quantities of Interest — Prefer Integral Over Local

Select the engineering quantities that matter for your application. Common choices include drag coefficient, lift coefficient, pressure drop, average Nusselt number, mass flow rate, or maximum temperature.

An important practical consideration: use integral (global) quantities whenever possible. A GCI based on the drag coefficient exercises the entire surface mesh and most of the volume mesh. A GCI based on the maximum velocity in a single cell tells you about the resolution at one point — and nothing about the rest of the domain. If your GCI study says the drag is well resolved, that does not mean the local heat transfer at some downstream location is equally converged.

Celik et al. (2008) make an additional point that is easy to miss: the definition of the representative cell size h should match the type of quantity being analyzed. The global average h = (V/N)^(1/D) — which this calculator uses by default — is appropriate for integral quantities like drag or pressure drop. For field variables at a specific location, the local cell size can be used instead. In practice, the global h is almost always the right choice for the quantities that matter most in engineering applications.

If you need assurance about local quantities, consider constructing integrated metrics: area-averaged temperature over a surface, velocity profiles integrated across a wake plane, or total heat flux through a boundary. These are more robust than single-point values and provide convergence information across larger mesh regions.

It is good practice to evaluate GCI for multiple quantities, since different quantities may converge at different rates. A mesh that is adequate for pressure drop might not be sufficient for local heat transfer. The calculator supports multi-quantity analysis — enter the grid parameters once and add as many quantities as you need.

Understanding the Results

Convergence Types

The calculator classifies your grid study into one of three convergence types based on the convergence ratio R = ε₃₂ / ε₂₁, where ε₃₂ = φ₃ − φ₂ and ε₂₁ = φ₂ − φ₁.

Monotonic convergence (0 < R < 1) means the solution changes consistently in the same direction as you refine the mesh, and the changes are getting smaller. This is the ideal case. The GCI is most reliable here, and the Richardson extrapolation provides a meaningful estimate of the grid-independent solution.

Oscillatory convergence (−1 < R < 0) means the solution alternates above and below the converged value with successive refinements. This can happen with non-uniform mesh refinement, when competing discretization errors partially cancel at certain mesh densities. The GCI can still be estimated, but with reduced confidence. If you encounter oscillatory convergence, consider adding a fourth mesh level or checking that your meshes are truly systematically refined.

Divergent behavior (|R| > 1) means the solution is getting worse — moving further from convergence — as you refine. This almost always indicates a problem outside the mesh itself: insufficient iteration convergence, inconsistent mesh topology between levels, a mesh-dependent boundary condition, or grids that are too coarse to be in the asymptotic range.

The Observed Order of Convergence (p)

Richardson extrapolation computes the apparent order of accuracy from your three solutions. For a second-order finite volume scheme, you would expect p ≈ 2. In practice, the observed order often differs from the formal order because of factors like non-uniform meshes, boundary effects, mixed-order limiters, and turbulence modeling.

An observed order between 1 and 3 is typical for well-behaved studies. When the observed order agrees well with the formal order, this is a good indication that the grids are in the asymptotic range. However, Celik et al. (2008) explicitly state that the converse — an observed order that deviates from the formal order — should not necessarily be taken as a sign of unsatisfactory calculations. Many factors can cause the observed order to differ from the nominal order, including non-uniform meshes, boundary treatment, mixed-order limiters, turbulence modeling, and the interplay between different discretized terms.

If p is very low (say p < 0.5), it may indicate that the meshes are not in the asymptotic range or that non-discretization errors are contaminating the results. If p is much higher than the formal order (say p > 5), the convergence may be coincidentally fast for this particular quantity, or the mesh differences are too small to properly resolve the trend. Unrealistic values of p combined with oscillatory convergence is a common sign of non-uniform refinement — check your meshing approach if you see this pattern.

The GCI Value

The GCI itself is reported as a percentage and represents an error band around your fine-grid solution. A GCI of 2 % means you can be reasonably confident that the true grid-independent solution lies within ±2 % of your fine-grid result.

As a rough guide: GCI below 2 % is generally considered excellent, 2–5 % is good for most engineering purposes and detailed studies, 5–10 % is acceptable for parametric studies where relative differences between configurations matter more than absolute values, and above 10 % suggests that further refinement is needed. The Mesh Sizing Estimator (Step 5 in the calculator) can help you determine how many cells you would need to reach a specific target.

The GCI uses a safety factor of Fs = 1.25 when three or more grids are used (the standard Celik et al. procedure). For two-grid studies, Fs = 3.0 is recommended to compensate for the inability to compute the observed order.

The Asymptotic Range Check

The asymptotic range check verifies that your grids are fine enough for the Richardson extrapolation theory to hold. It is computed as GCI₃₂ / (r₂₁ᵖ × GCI₂₁). A value close to 1.0 confirms that the grids are in the asymptotic range — meaning the leading-order error term dominates and the extrapolation is reliable.

Values significantly different from 1.0 (say below 0.8 or above 1.2) suggest that either the coarsest grid is too coarse, the refinement is not truly systematic, or higher-order error terms are still significant. In such cases, the GCI should be treated as a rough estimate rather than a tight bound.

Estimating Required Cells for a Target GCI

Once you have completed a grid study, a natural question is: “How many cells would I need to bring the GCI below a certain threshold?” The Mesh Sizing Estimator in the calculator answers this by reversing the GCI relationship.

The key formula is:

h_new / h_fine = (GCI_target / GCI_fine)^(1/p)

From this new representative cell size, the estimated cell count follows:

N_new = N_fine / (h_new / h_fine)^D

For example, if your current fine grid has 500,000 cells with GCI = 5 % and observed order p = 2 (3D simulation), achieving GCI = 1 % would require approximately 5.6 million cells — a factor of 11× increase. This steep scaling is a direct consequence of the GCI formula and illustrates why it is often not economical to push for extremely low GCI values when the current level is already acceptable for the application.

The estimator also identifies quantities that are already over-resolved relative to your target, which can reveal opportunities to use a coarser mesh and save computational cost.

Practical Tips and Common Pitfalls

When φ Is Close to Zero

Some quantities of interest — such as the lift coefficient at zero angle of attack — may have values near zero. This causes the relative error to explode even when the absolute changes between meshes are tiny. The calculator provides a “Ref. Value” field for this situation. Enter a characteristic scale for the quantity, such as the corresponding drag coefficient for a near-zero lift coefficient, or the maximum value of the quantity in the domain. This normalizes the GCI against a meaningful reference instead of the near-zero fine-grid value.

Iteration Convergence Must Come First

A mesh convergence study only makes sense if each individual simulation is converged to a sufficient level. Celik et al. (2008) are specific about this: normalized residuals should decrease by at least three orders of magnitude, and preferably four, before results are extracted. The iteration error should be at least one order of magnitude smaller than the discretization error you are trying to measure — otherwise you are quantifying iterative noise, not mesh sensitivity.

It is also worth emphasizing: residuals dropping below a convergence threshold does not guarantee that your solution is converged. The residuals only measure how much the solution changes between iterations, and aggressive under-relaxation can make them artificially small. Monitor the actual quantities of interest (drag, pressure drop, temperature) and verify that they have stabilized. Taking an average over the last several hundred iterations, rather than the value at the final iteration, gives a more reliable measurement.

The Celik et al. paper also outlines an eigenvalue-based method (Appendix A) for estimating iteration error more rigorously. While this is rarely done in practice, it provides a formal framework when the standard residual-based approach is insufficient.

Blended and Hybrid Schemes — A Common Trap

The ASME Journal of Fluids Engineering editorial policy requires that numerical methods be at least formally second-order accurate in space for archival publication. This is not merely a preference — first-order methods are explicitly excluded because their numerical diffusion has been shown to devastate solution accuracy. The policy also recognizes that higher-order methods, while more expensive per grid point, are far more efficient in terms of accuracy per total computational cost.

The policy further states that methods using blending or switching between first and second order — including the well-known hybrid upwind, power-law, and exponential schemes — are to be treated as first-order methods for the purpose of GCI calculations, unless it can be demonstrated that their numerical diffusion does not dominate the physical diffusion. A similar policy applies to schemes with significant amounts of explicitly added artificial viscosity.

This matters because some CFD codes activate blending or limiters by default. If your solver blends between first-order upwind and second-order central differencing based on a local Peclet number, ASME would consider this a first-order scheme. In that case, set the nominal order in the calculator to 1, not 2.

If you observe an apparent order of convergence much lower than 2 despite using what you believe is a second-order scheme, this is one possible explanation. Check your solver’s discretization settings carefully — particularly for the convective terms, which are the most affected by numerical diffusion. Note, however, that the Celik et al. procedure explicitly states that a mismatch between the observed order and the formal order should not necessarily be taken as a sign of unsatisfactory calculations. Many real-world simulations produce observed orders between 1 and 2 on unstructured meshes even with formally second-order schemes, due to mesh non-uniformity, boundary treatment, and limiter effects.

Transient Simulations Need Separate Studies

The GCI method applies strictly to spatial discretization. For time-dependent simulations, you need separate convergence studies for the time step and the mesh. First establish a sufficiently small time step on a given mesh, then perform the spatial GCI study with that time step held constant. Changing both simultaneously makes it impossible to attribute the solution changes to mesh refinement alone.

The ASME editorial policy additionally requires that temporal accuracy be demonstrated so that spurious effects of phase error are shown to be limited. In particular, unphysical oscillations due to numerical dispersion should be significantly smaller in amplitude than the captured short-wavelength features of the flow. For transient cases, also verify that iterative convergence is achieved at every time step, not just checked at a single snapshot.

Structured vs. Unstructured Meshes

The GCI method was originally developed for structured grids with uniform refinement, where the cell size is well-defined and consistent everywhere. For unstructured meshes (polyhedral, tetrahedral, or trimmed cells), the representative cell size h = (V/N)^(1/D) is an average measure that does not capture local variations in mesh density.

Celik et al. (2008) emphasize that “the refinement itself should be structured even if the grid is unstructured” and that “use of geometrically similar cells is preferable.” In practice, this means changing only the global base size parameter between mesh levels, keeping all relative sizing, curvature controls, and growth rates identical. The mesher should produce topologically similar meshes at each level — just with more cells.

If you see unexpected results (oscillatory convergence, unrealistic p), consider whether your mesh topology is changing significantly between refinement levels. Unstructured meshers can produce qualitatively different meshes at different base sizes — different cell types in certain regions, different transition patterns between refinement zones — which undermines the systematic refinement assumption.

Inlet Boundary Condition Sensitivity

An often-overlooked source of uncertainty is the sensitivity of your results to inlet boundary conditions. Celik et al. (2008) recommend studying and reporting the degree of sensitivity of the solution to small perturbations in the inlet conditions. This is separate from the mesh convergence study, but equally important for establishing confidence in the results.

The ASME editorial policy also emphasizes that the overall accuracy of a simulation is strongly affected by the implementation and order of the boundary conditions. The boundary condition treatment may have a different order of accuracy than the interior scheme, and this can become the dominant source of error if not handled carefully. When appropriate, particular attention should be paid to inflow and outflow boundary conditions.

For example, if your drag coefficient changes by 5 % when you perturb the inlet turbulence intensity by 10 %, that 5 % uncertainty exists on top of whatever GCI value you compute. The GCI only quantifies the mesh-induced component; it says nothing about how uncertain the boundary conditions themselves are.

Reporting Discretization Error Bars on Profiles

When presenting velocity, temperature, or pressure profiles in publications, Celik et al. (2008) recommend adding discretization error bars analogous to experimental uncertainty bars. The procedure uses the GCI formula applied at each point along the profile, but with a global average observed order (p_ave) rather than the local p at each point.

The local p can vary dramatically — in the backward-facing-step example from the Celik paper, it ranges from 0.012 to 8.47 along a single velocity profile, while the global average is p_ave = 1.49. Using the local p would produce wildly inconsistent error bars; the averaged value gives a much more physically meaningful result.

This approach is straightforward to implement: extract the quantity at each profile point for all three grids, compute the GCI at each point using p_ave and the local fine-grid value, and plot the result as ± error bars. Our calculator handles integral quantities; for profile error bars, use the observed order from an integral-quantity GCI study as your p_ave.

Noisy Grid Convergence and Least-Squares GCI

In real-world applications, grid convergence is not always clean. Some quantities may show slight irregularities due to mesh topology changes, non-uniform refinement, or competing error terms. For such cases, the Celik et al. paper references the least-squares version of GCI developed by Eça and Hoekstra as an alternative approach.

The least-squares method uses more than three grids (typically five or more) and fits the Richardson extrapolation model to the data using a least-squares procedure. This is more robust to individual outlier grids and provides statistical confidence bounds on the extrapolated value. If you have access to more than three mesh levels and observe noisy convergence behavior, this approach may give more reliable uncertainty estimates than the standard three-grid procedure.

When GCI Adds Less Value

If you have high-quality experimental data for the exact configuration you are simulating, comparing directly against that data may be more informative than a GCI study alone. The GCI tells you about numerical convergence; the experimental comparison tells you about physical accuracy. In an ideal workflow, you would do both: use GCI to verify that your results are mesh-independent, and then validate against experimental data to confirm accuracy.

The ASME editorial policy makes two important caveats about validation. First, when comparing with experimental results, the experimental uncertainty must be established — “reasonable agreement” alone is not sufficient to justify a single-grid calculation, especially when adjustable parameters such as turbulence model coefficients are involved. Second, validation against analytical solutions or well-established benchmarks demonstrates accuracy only for that particular class of problems. It does not automatically extend to other problem types, particularly when adjustable modeling parameters are involved. A simulation that matches a canonical backward-facing-step benchmark beautifully may still fail for a complex industrial geometry.

Similarly, if established best practices exist for your application — including recommended mesh sizes, y⁺ targets, and refinement zones — following those guidelines and validating against reference data may be more efficient than running a full GCI study from scratch. The GCI is most valuable when you are working in new territory without existing guidelines to rely on.

Using This Calculator with Simcenter STAR-CCM+

Finding the cell count: Right-click on the mesh region and select Mesh Diagnostics, or create a Volume Mesh Cell Count report under Reports. The latter approach is more convenient when comparing multiple meshes.

Systematic refinement: Adjust the Base Size under the mesh continuum. Use a constant ratio between levels — for example, if your fine mesh uses a base size of 2 mm, the medium mesh should use 2 × √2 ≈ 2.83 mm, and the coarse mesh 2 × 2 = 4 mm. This gives refinement ratios of approximately 1.41 between each level. Keep all other mesh settings identical: same Prism Layer count, stretching, and total thickness; same surface sizes and curvature controls.

Extracting quantities: Create Reports for each quantity of interest (drag force, pressure drop, area-averaged temperature, etc.). Run all three meshes to convergence and enter the report values in the calculator. The multi-quantity feature lets you analyze several quantities at once — the grid data stays the same, only the φ values change.

Domain volume (optional): If you want the calculator to display absolute cell sizes rather than just ratios, create a Volume Report on your fluid region, or use Derived Parts → Sum of Cell Volumes. This field is optional — relative refinement ratios from cell counts are sufficient for the GCI calculation.

Using the Mesh Sizing Estimator: After computing GCI for your quantities, enter a target GCI percentage in Step 5. The estimator will tell you how many cells you would need — or whether your current mesh is already finer than necessary. This is particularly useful when planning computational budgets for parametric studies.

The Mathematics Behind GCI

For those who want to understand the underlying method, here is a brief summary. The GCI is based on Richardson extrapolation, which assumes that the discretization error can be expressed as a power series in the cell size h.

Given three solutions φ₁, φ₂, φ₃ on grids with representative sizes h₁ < h₂ < h₃, the apparent order of convergence p is found by solving:

p = ln(|ε₃₂ / ε₂₁|) / ln(r₂₁)

where ε₃₂ = φ₃ − φ₂, ε₂₁ = φ₂ − φ₁, and r₂₁ = h₂/h₁. For non-uniform refinement ratios (r₂₁ ≠ r₃₂), the equation becomes implicit and is solved iteratively — the calculator handles this automatically using the Celik et al. (2008) procedure.

The Richardson-extrapolated (grid-independent) solution from the fine and medium grids is:

φ_ext²¹ = (r₂₁ᵖ · φ₁ − φ₂) / (r₂₁ᵖ − 1)

Celik et al. (2008) also require computing the corresponding extrapolation from the medium and coarse grids:

φ_ext³² = (r₃₂ᵖ · φ₂ − φ₃) / (r₃₂ᵖ − 1)

Comparing these two extrapolated values provides an additional consistency check — if they differ significantly, it may indicate that the coarser grids are not yet in the asymptotic range.

And the Grid Convergence Index for the fine grid is:

GCI_fine = Fs · |ε₂₁ / φ₁| / (r₂₁ᵖ − 1)

with the safety factor Fs = 1.25 for the three-grid procedure.

The Mesh Sizing Estimator reverses this relationship. Since GCI scales approximately as hᵖ, the required refinement to achieve a target GCI is:

h_new / h_fine = (GCI_target / GCI_fine)^(1/p)

From which the estimated cell count is N_new = N_fine / (h_new / h_fine)^D.

References and Further Reading

The formulas in this calculator follow the procedure described in Celik et al. (2008), “Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications,” ASME Journal of Fluids Engineering, 130(7):078001. The original GCI concept was introduced in Roache (1994), “Perspective: A Method for Uniform Reporting of Grid Refinement Studies,” Journal of Fluids Engineering, 116(3):405–413. The ASME editorial policy on numerical accuracy (Roache et al. 1986, revised by Freitas 1993 and Celik et al. 2008) provides the broader framework within which the GCI procedure sits.

For noisy grid convergence, the least-squares GCI approach by Eça and Hoekstra (2005, 2007) provides a more robust alternative using five or more grids. For a comprehensive treatment of verification and validation in CFD, see Roache (1998), Verification and Validation in Computational Science and Engineering, Hermosa Publishers, and Oberkampf and Roy (2010), Verification and Validation in Scientific Computing, Cambridge University Press.

Looking for more CFD tools? Try our Turbulence Inlet BC Calculator, Y⁺ Calculator, Nusselt Number Calculator, or Heat Transfer Coefficient Correlations.