---
tag: X001F42KH8
kind: concept
name: Adaptive mesh refinement
msc: [65N50, 65M50]
sources:
  - url: https://en.wikipedia.org/wiki/Adaptive_mesh_refinement
same_as:
  wikidata: Q4680751
  wikipedia: Adaptive mesh refinement
---

## Summary

Adaptive mesh refinement makes a computational mesh finer only where the solution needs it, guided by the computed solution itself.
In [[finite element methods|98GZ9DVHTA]] it runs a loop: solve, estimate the error on each element, mark elements with large error, and refine them.
For time-dependent hyperbolic problems, block-structured refinement in space and time was developed by Marsha Berger and Joseph Oliger.
Adaptivity recovers optimal convergence rates for solutions with singularities, where uniform refinement is slow.
See the [[definition|C5JM6D0841]].

## Explanations

### L1 human

A uniform mesh spends as much effort on calm regions as on regions where the solution changes sharply.
Adaptive refinement looks at the current approximation and adds detail only where the error seems large.
Near a sharp corner of a domain, or near a shock wave, the mesh becomes very fine, while elsewhere it stays coarse.
For the Poisson equation on an L-shaped domain, the elements cluster around the inner corner, where the solution is singular.