Convergence
Mesh convergence and solver convergence — how to verify your FEM result is trustworthy.
Convergence is how you know a FEM result can be trusted. There are two distinct types: mesh convergence (does refining the mesh still change the result?) and solver convergence (did the iterative solver actually find equilibrium?).
Mesh convergence
FEM is an approximation. The coarser the mesh, the further the result is from the exact solution. As the mesh is refined, the FEM result should approach the true answer — this is mesh convergence.
How to perform a convergence study
- Start with a coarse mesh and record the key result (peak stress, max displacement)
- Refine the mesh (halve the element size) and record the result again
- Repeat until the result changes by less than 5%
- That is your converged result
You don't need to refine the mesh globally. Refine only in the regions you care about — typically where stress is highest or where the geometry has the smallest features.
What convergence looks like
| Mesh size | Peak stress (MPa) | | --- | --- | | Coarse (5 mm) | 182 | | Medium (2.5 mm) | 203 | | Fine (1.25 mm) | 208 | | Very fine (0.6 mm) | 209 |
Between "fine" and "very fine", the result changes by less than 0.5%. The result is converged at the fine mesh.
What non-convergence looks like
If peak stress keeps growing without a limit as the mesh is refined, you have a singularity (see the Singularities article). That stress value is not physical — it will never converge.
Solver convergence
For nonlinear analyses (contact, plasticity, large deformation), the solver iterates to find equilibrium. Each iteration reduces the residual — the out-of-balance force between internal and external loads. The solver converges when the residual drops below a tolerance.
Signs of solver divergence
- Residual increases from one iteration to the next
- Solver terminates early with "convergence not achieved"
- Unrealistically large displacements in the result
Common fixes
- Reduce the load step size (apply the load more gradually)
- Refine the mesh in the contact or high-plasticity regions
- Improve contact geometry (reduce large gaps, clean sharp edges)
- Tighten material model parameters (check yield stress and hardening)
A nonlinear analysis that reports "converged" is not necessarily correct. Check that the residual is genuinely below the tolerance and that the deformed shape makes physical sense.
The 5% rule
A commonly used practical standard: the result is considered converged when two successive mesh refinements give results within 5% of each other. This is not a formal mathematical criterion — it's engineering judgement. For safety-critical applications, use a tighter tolerance (1–2%).