| Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. This talk will describe a new adaptive method based on a subdivision strategy known as newest vertex bisection, and approximation by piecewise linear elements. It is proven that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the $H^1$ norm). Namely, it is shown that whenever $s>0$ and the solution $u$ is such that for each $n\ge 1$, it can be approximated to accuracy $O(n^{-s})$ in the energy norm by a continuous, piecewise linear function on a triangulation with $n$ cells (using complete knowledge of $u$), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order $O(n)$ for each $n\ge 1$. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation. |