Mesh Quality and Resolution
The discretization mesh shapes the solution's accuracy; poor quality or resolution corrupts results before any equation is solved.
The Grid Beneath Everything
Before any equation is solved, the continuous domain is divided into a mesh of cells or elements. The quality and resolution of that mesh set an upper bound on achievable accuracy. A mesh too coarse to resolve a feature, or made of badly shaped cells, corrupts the solution no matter how good the solver, and it undermines the convergence studies that verification depends on.
Quality Measures
- Cell shape: highly distorted, skewed, or high-aspect-ratio cells degrade accuracy and can destabilize solvers.
- Smoothness: abrupt changes in cell size between neighbors introduce error at the transition.
- Orthogonality: for some methods, non-orthogonal cells at boundaries reduce accuracy.
- Resolution: enough cells to capture the gradients and features of the solution.
Resolving the Physics
Adequate resolution is problem-dependent: a region of steep gradients, a boundary layer, or a small geometric feature needs finer cells than a smooth interior. Under-resolving such regions produces a solution that looks smooth but is wrong, because the feature driving the physics is smeared out. Adaptive refinement, which adds cells where the solution varies most, targets resolution where it is needed without refining everywhere.
Interaction with Verification
Mesh quality directly affects convergence studies. A refinement sequence must keep the mesh family self-similar and well shaped; if refinement introduces distorted cells or changes cell shape inconsistently, the observed order of accuracy is corrupted and the error estimate is unreliable. A clean convergence study requires a systematically refined, good-quality mesh family, which is why mesh generation is part of the verification workflow, not a preliminary to be rushed.
The practical rule: a result on a mesh that has not been shown adequate through a resolution study is provisional. The mesh is a modeling choice with its own error contribution, and that contribution belongs in the error budget alongside the scheme's discretization error, which it can dominate when resolution is marginal.