Ron DeVore

Adaptive Finite Element Methods

Coauthors: Peter Binev and Wolfgang Dahmen
Adaptive Finite Element Methods (AFEM) are frequently used in the numerical solution of PDEs. However, for elliptic equations, it is only recently that these methods have been shown to converge. We will go much further and actually construct an AFEM which has optimal convergence rates in terms of error decay versus number of computations. We shall concentrate on two results that have proven to be important in the proof of these convergence rates. The first is how to bound the number of additional subdivisions needed to remove hanging nodes in adaptive methods. The second result is how to generate a near best adaptive approximation to a given function in linear time.