This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators.Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

A mathematical model for a finite-strain elastoplastic evolution problem is proposed in which one time-step of an implicit time-discretization leads to generally non-convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip-system and von Mises plasticity illustrate that finite-strain elastoplasticity generates reduced problems with non-quasiconvex energy densities and so allows for non-attainment of energy minimizers and microstructures.

Abstract. A computable error bound for mixed finite element methods is established in the model case of the Poisson-problem to control the error in the H(div,Ω) ×L 2 (Ω)-norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart-Thomas, Brezzi-Douglas-Marini, and Brezzi-DouglasFortin-Marini elements.