Error estimates and adaptivity

Abstract

In this section, we come to further applications of approximation and function space theory to numerical multilevel schemes. Traditionally, we need them to justify the discretization process and to obtain qualitatively and quantitatively correct convergence estimates. In addition to the usual a priori error estimates (see briefly in 5.1), which depend on different regularity assumptions on the exact solution of the variational problem under consideration, one is equally interested in computable a posteriori estimates. The latter are important for adaptive processes (feedback control, as an example one can think about dynamic grid generation or refinement design). If such estimators are reliable and reflect the actual error (not the theoretical asymptotic upper bound which is typical for a priori error estimates) they may be of real importance for engineering applications. There is a huge volume of recent research literature about this topic, and we restrict our attention to a very particular but promising direction which was the starting point for us when getting involved in multilevel finite element approximation [Os1]. Such bounds can be treated within the framework of the approximation spaces A_p,qs ({V _j }) introduced in 3.4. In 5.2 we prove some theoretical results on subspace selection related to the h-version of the finite element method (local grid refinement). The possible algorithmical consequences are discussed in 5.3. Another possibility to design an adaptive method based on the stable BPX-splittings of 4.2.1-2 is briefly mentioned in 5.3.