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We consider an adaptive finite element method (AFEM) for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to $R$-linear convergence. This result is shown to hold up to a consistery error due to the extension of the discrete multipliers (point functionals) to $H^{-1}$ and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.
A refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first-order adaptive finite element method (AFEM) is presented. The $H^1$ stability of the $L^2$ projection provides reliability and efficiency of the edge-contribution of standard residual-based error estimators for $P_1$ finite element methods. In fact, the volume contributions and even oscillations can be omitted for Courant finite element methods. This allows for a refined averaging scheme and so improves [Dong Mao, Lihua Shen and Aihui Zhou, Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates, Advanced in Computational Mathematics, 2006, 25: 135-160]. The proposed AFEM
monitors the edge-contributions in a bulk criterion and so enables a contraction property up to higher-order terms and global convergence. Numerical experiments exploit the remaining $L^2$ error contributions and confirm our theoretical findings. The averaging schemes show a high accuracy and the AFEM leads to optimal empirical convergence rates.
The discontinuous Galerkin (dG) method provides a hierarchy of time discretization schemes for evolutionary problems. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in [Alberty, Carstensen: Discontinuous {G}alerkin Time Discretization in Elastoplasticity: Motivation, Numerical Algorithms and Applications, Comp. Meth. Appl. Mech. Engrg. \textbf{191} (2002)] with the help of duality in convex analysis to justify certain jump terms. Convincing numerical experiments have already been displayed in the literature.\This paper establishes a mathematical a~priori error analysis for the dG($1$) scheme with discontinuous piecewise linear polynomials in the temporal and first-order finite elements for the spacial discretization. One novel key idea in the a~priori convergence analysis is an optimal trace estimate under convex constraints. The numerical investigation of the empirical convergence rate in a benchmark concludes the paper.