Equidistribution and Optimal Approximation Class

Local adaptive grid refinement is an important technique in finite element methods. Its study can be traced back to the pioneering work [2] in one dimension. In recent years, mathematicians start to prove the convergence and optimal complexity of the adaptive procedure in multi-dimensions. [11] first proved an error reduction in the energy norm for the Poisson equation provided the initial mesh is fine enough. [15, 16] extended the convergence result without the constrain of the initial mesh and they also reveal the importance of data oscillation. But results in [11, 15, 16] only establish the qualitative convergence estimate by a proof of an error reduction property. The number of elements generated by the adaptive algorithm is not under control. A natural theoretical question is if a standard adaptive finite element scheme would give an optimal asymptotic convergence rate in terms of the number of elements. For linear finite element approximation to second order elliptic boundary value problems in two dimensions, for example, an optimal asymptotic error estimate would be something like |u uN |1,W C(u)N 1/2, (1)