Adaptive numerical methods for PDEs

Abstract
While adaptive numerical methods are often used in solving partial differential equations, there is not yet a cohesive theory which justifies their use or analyzes their performance. The purpose of this talk is to put forward the first building blocks of such a theory, the cornerstones of which are nonlinear approximation and regularity theorems for PDEs. Any adaptive numerical method can be viewed as a form of nonlinear approximation: the solution u of the PDE is approximated by elements from a nonlinear manifold of functions. The theory of nonlinear approximation relates the efficiency of this type of approximation to the regularity of u in a certain family of Besov spaces. Regularity for PDEs are needed to determine the smoothness of u in this new Besov scale. Together, the approximation theory and regularity theory determine the efficiency of approximation that is possible using adaptive methods. A similar analysis gives the efficiency of linear algorithms. The two can then be compared to predict whether nonlinear methods would result in better performance. Examples will be given in the setting of both elliptic and hyperbolic problems. A wavelet based algorithm for elliptic equations developed by Albert Cohen, Wolfgang Dahmen, and the author will be presented as one of the successes of this theory.