Ronald A. DeVore
| Adaptive methods are often used to numerically resolve PDEs when the solution to the PDE is known to exhibit singularities. Yet, it is rare that there is even a convergence theory for an adaptive numerical method much less an analysis of the decay of error in terms of the number of computations. In this talk, we shall develop an analysis which will allow the a priori determination of whether an adaptive numerical method can perform better than the more standard (and numerically less intensive) linear methods such as standard finite element methods. Since adaptive methods are a form of nonlinear approximation, it is not surprising that this theory has as one of its pillars the fundamental theorems which characterize approximation rates (in terms of smoothness conditions on the target function) of nonlinear methods. The other pillar for this theory is regularity of the solution to the PDE. But the new twist is that the regularity is not measured in the usual Sobolev scale but rather in a scale of Besov spaces commensurate with nonlinear methods. We shall give examples of how to apply this theory to hyperbolic and elliptic problems. |