Error Estimation and Adaptivity
How solvers measure their own error and refine resolution where it is needed, delivering accuracy without uniform over-resolution.
Computing where you need it
Uniformly fine resolution wastes work where the solution is smooth and may still under-resolve localized features like boundary layers, shocks, or steep gradients. Adaptivity refines resolution only where an error estimate says it is needed. This requires two ingredients: a reliable estimate of the local error, and a strategy for acting on it.
A posteriori error estimation
An a posteriori error estimate uses the computed solution itself to bound or approximate the error. Common approaches include comparing solutions at two resolutions or two method orders (embedded Runge-Kutta pairs do this in time), evaluating the residual of the discrete solution in the governing equation, and recovery-based estimators that compare a smoothed gradient to the raw one. Goal-oriented estimators, built on the adjoint method, target the error in a specific quantity of interest rather than a global norm.
Acting on the estimate
- h-refinement: subdivide cells or shrink the time step where error is large
- p-refinement: raise the polynomial order locally where the solution is smooth
- r-refinement: move existing nodes toward regions needing resolution
- hp-adaptivity: combine h and p to achieve exponential convergence
The adaptive loop
The canonical strategy is solve, estimate, mark, refine, repeat: solve the discrete problem, estimate the error per element, mark the elements with the largest error, refine them, and re-solve. Time integrators do the same in one dimension, accepting or rejecting steps and adjusting the step size to keep the local error near a target. Well-designed adaptivity provably converges toward the true solution while keeping the number of unknowns close to optimal.
Reliability and efficiency
A good estimator is both reliable (it does not underestimate the true error, so refinement is triggered when needed) and efficient (it does not grossly overestimate, so work is not wasted). Balancing these is the core of adaptive-method theory. Adaptivity is essential in fusion modeling for resolving steep pedestal gradients and thin current layers without paying for fine resolution everywhere.