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  "body": "\n## Summary\n\nAdaptive mesh refinement makes a computational mesh finer only where the solution needs it, guided by the computed solution itself.\nIn [[finite element methods|98GZ9DVHTA]] it runs a loop: solve, estimate the error on each element, mark elements with large error, and refine them.\nFor time-dependent hyperbolic problems, block-structured refinement in space and time was developed by Marsha Berger and Joseph Oliger.\nAdaptivity recovers optimal convergence rates for solutions with singularities, where uniform refinement is slow.\nSee the [[definition|C5JM6D0841]].\n\n## Explanations\n\n### L1 human\n\nA uniform mesh spends as much effort on calm regions as on regions where the solution changes sharply.\nAdaptive refinement looks at the current approximation and adds detail only where the error seems large.\nNear a sharp corner of a domain, or near a shock wave, the mesh becomes very fine, while elsewhere it stays coarse.\nFor the Poisson equation on an L-shaped domain, the elements cluster around the inner corner, where the solution is singular.\n",
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