In Sect. 0.8, we demonstrated the possibility of dramatic improvements in approximation power resulting from adaptive meshes. In current computer simulations, meshes are often adapted to the solution either using a priori information regarding the problem being solved or a posteriori after an initial attempt at solution (Babuška et al. 1983 & 1986). The resulting meshes tend to be strongly graded in many important cases, no longer being simply modeled as quasi-uniform. Here we present some basic estimates that show that such meshes can be effective in approximating difficult problems. For further references, see (Eriksson, Estep, Hansbo and Johnson 1995), (Verfürth 1996), (Ainsworth and Oden 2000), (Becker and Rannacher 2001), (Babuška and Strouboulis 2001) and (Dörfler and Nochetto 2001).
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