# Convergence Analysis of Adaptive Non-Standard Finite Element Methods

## Ronald H.W. Hoppe

**Oct. 17, 2008, 11 a.m. HF 136**

Adaptive ﬁnite element methods have become powerful tools for the eﬃcient and reliable numerical solution of partial diﬀerential equations and systems thereof. They consist of successive loops of the cycle

SOLVE ⇒ ESTIMATE ⇒ MARK ⇒ REFINE .

Here, SOLVE stands for the solution of the ﬁnite element discretized problem with respect to a given triangulation of the computational domain using, e.g., advanced iterative solvers based on multilevel and/or domain decomposition methods. The following step ESTIMATE provides a cheaply computable, localizable a posteriori error estimator for the global discretization error or some other problem-specic quantity of interest. The subsequent step MARK deals with the selection of elements, faces and/or edges of the triangulation for reﬁnement and/or coarsening, whereas the ﬁnal step REFINE takes care of the technical realization of the reﬁnement/coarsening process.

An important issue is the convergence analysis of the adaptive loop in the sense of a guaranteed reduction of the underlying error functional. During the past decade, such a convergence analysis has been successfully established mainly for standard conforming ﬁnite element discretizations of second order elliptic boundary value problems. In this contribution, we focus on recent results for non-standard discretizations such as mixed and mixed-hybrid methods as well as nonconforming techniques including Discontinuous Galerkin methods.