A computational technique that dynamically refines and coarsens a discretization mesh during simulation based on local error indicators or feature detection so that resolution is concentrated where it is most needed.

Principle

Use a posteriori error estimates or feature detectors to decide where mesh elements should be subdivided or merged, balancing accuracy and computational cost by increasing local degrees of freedom only in regions with large errors or complex solution structure.

Demonstration

In compressible fluid dynamics, AMR refines the grid around shock fronts and contact discontinuities while coarsening in smooth flow regions, enabling accurate capture of discontinuities without uniformly fine meshes across the entire domain.

Misapplication

Refining based on noisy or poorly chosen indicators can produce oscillatory refinement/coarsening cycles, excessive refinement in transient artifacts, or violation of conservation when interpolation between levels is non-conservative.

Consequence

Proper AMR reduces computational cost for a target accuracy, enables localized high-resolution features and adaptive time-stepping, but requires careful data structures, error control, and load balancing in parallel implementations.

Reversal

Uniform mesh refinement increases resolution everywhere and guarantees predictable discretization error reduction but at much higher cost; disabling adaptivity trades efficiency for simplicity and often impractical resource use for large-scale problems.

Boundary

Applies to discretizations (finite difference, finite volume, finite element) where locality of error can be estimated; it excludes problems where global coupling prevents meaningful local refinement or where solution features are uniformly small-scale everywhere.

Semantic Tension

Tension with hp-adaptivity and multiresolution methods: AMR changes mesh resolution (h), hp-adaptivity additionally varies polynomial order (p) on elements, and multiresolution methods represent scales hierarchically; they can be combined but entail different implementation and error-control strategies.

Synthesis

Adaptive Mesh Refinement concentrates computational resources by dynamically modifying mesh resolution according to local error and feature detection, achieving targeted accuracy with reduced cost while introducing algorithmic complexity for interpolation, conservation, and parallel scaling.