Adaptive numerical methods for PDEs

Abstract

While adaptive numerical methodsare often used in solving partial differential equations, there isnot yet a cohesive theory which justifies their useage or analyzestheir performance. The purpose of this talk is to put forward thefirst building blocks of such a theory, the cornerstones of which arenonlinear approximation and regularity theorems for PDEs. Any adaptivenumerical method can be viewed as a form of nonlinear approximation: thesolution u of the PDE is approximated by elements from a nonlinearmanifold of functions. The theory of nonlinear approximation relatesthe efficiency of this type of approximation to the regularity ofu in a certain family of Besov spaces. Regularity for {{}PDE{}}s areneeded to determine the smoothness of u in this new Besov scale.Together, the approximation theory and regularity theory determine theefficiency of approximation that is possible using adaptive methods.A similar analysis gives the efficiency of linear algorithms. Thetwo can then be compared to predict whether nonlinear methods wouldresult in better performance. Examples will be given in the setting ofboth elliptic and hyperbolic problems. A wavelet based algorithm forelliptic equations developed by Albert Cohen, Wolfgang Dahmen, and theauthor will be presented as one of the successes of this theory.