A numerical technique that dynamically refines and coarsens a computational mesh based on local error indicators or solution features so as to concentrate degrees of freedom where higher resolution is needed.

Principle

Local refinement decisions are driven by a posteriori error estimates, gradient or curvature indicators, or feature detectors; meshes are hierarchically adapted to balance accuracy and computational cost.

Demonstration

In a compressible flow simulation refine the mesh locally around shock fronts and contact discontinuities based on large solution gradients, while coarsening in smooth regions to save resources.

Misapplication

Refining solely by heuristic without error control can create hanging nodes or violate conservation laws under improper projection between levels, and over-refinement may amplify numerical instability.

Consequence

Delivers target accuracy with significantly fewer global degrees of freedom than uniform refinement, enabling tractable resolution of localized phenomena in large-scale simulations.

Reversal

Uniform mesh refinement where resolution is increased globally irrespective of local solution behavior, often producing excessive cost for modest accuracy gains.

Boundary

Applies to discretizations that admit local mesh modification (finite element, finite volume, finite difference on adaptive grids); excludes methods that require fixed global bases without local adaptivity.

Semantic Tension

h-refinement (local element subdivision) versus p-refinement (increasing local approximation order): both aim to reduce discretization error but use different mechanisms and error estimators.

Synthesis

Adaptive mesh refinement is a hierarchical mesh strategy that uses local error or feature indicators to selectively refine and coarsen the computational grid, concentrating resources where solution complexity demands finer resolution.