ABSTRACT
In most numerical procedures for solving PDEs, the problem is first discretized by choosing algebraic equations on a finite-dimension approximation space. A numerical process is then devised to solve this huge system of discrete equations. The discretization process, which is unable to predict the proper resolution and the proper order of approximation at each location, produces a grid that is too fine. The algebraic system thus becomes unnecessarily large in size, while accuracy usually remains rather low (Brandt, 1977). The aim of adaptive methods is the generation of a grid that is adapted to the problem so that a given error criterion is fulfilled by the solution on this grid. An optimal grid should be as coarse as possible while meeting the criterion in order to save on computing time and memory requirements (Schmidt and Siebert, 2005). For stationary issues, a grid is almost optimal when the local errors are approximately equal for all elements. Therefore, elements where the error is large will be marked for refinement, while elements with small estimated errors are left unchanged or marked for coarsening.
