Mesh Adaptation and Error Estimation
Error estimators measure where a discrete solution is least accurate, driving adaptation that puts resolution where it reduces error most.
Letting the error guide the mesh
A uniform mesh spends effort evenly, but discretization error is almost never uniform: it concentrates where the solution varies rapidly or is singular. Mesh adaptation reallocates resolution toward those regions, guided by an error estimator that assigns each element an indicator of its contribution to the total error. Refining high-error elements and coarsening low-error ones drives the mesh toward an efficient distribution for a target accuracy.
Adaptation may change element size (h-adaptivity), polynomial degree (p-adaptivity), or element shape and orientation (r- and anisotropic adaptivity), or combine them.
A posteriori error estimators
A posteriori estimators compute error indicators from the numerical solution itself. Residual-based estimators measure how badly the discrete solution fails to satisfy the equation locally, including jumps in fluxes across element faces. Recovery-based estimators (such as Zienkiewicz-Zhu) compare the raw gradient with a smoothed, recovered gradient; their difference estimates the error. Goal-oriented (dual-weighted-residual) estimators target the error in a specific output quantity, weighting local residuals by the sensitivity of that output.
The adaptive loop
Adaptation runs as a loop: solve, estimate error, mark elements exceeding a threshold, refine or coarsen, and solve again. The loop continues until the estimated error falls below tolerance. Goal-oriented adaptation is especially valuable when a single engineering quantity, rather than the whole field, must be computed accurately and efficiently.
- Error indicators rank elements by their error contribution
- Residual, recovery, and goal-oriented estimators
- h-, p-, r-, and anisotropic adaptation strategies
- Solve-estimate-mark-refine loop to a tolerance
Error-driven adaptation lets fusion simulations resolve thin, moving features, such as steep edge gradients or current layers, accurately while keeping the overall element count and cost in check.