When choosing a numerical method to approximate the solution of a continuous mathematical problem, we need to consider which method results in an approximation that is not only close to the solution of the original problem, but possesses the important qualitative properties of the original problem, too. For linear elliptic problems the main qualitative properties are the various maximum principles. The preservation of the weak maximum principle was extensively investigated in the last decades, but not the strong maximum principle preservation. In this paper we focus on the latter property by giving its necessary and sufficient conditions, investigating the relation of the preservation of the strong and weak maximum principles and illustrating the differences between them with numerous examples. © 2012 Springer-Verlag.

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. We will demonstrate the performance and the favor of the method through numerical experiments. © 2012 Versita Warsaw and Springer-Verlag Wien.