In this paper we describe PDELab, an extensible C++ template library for finite element methods based on the Distributed and Unified Numerics Environment (Dune). PDELab considerably simplifies the implementation of discretization schemes for systems of partial differential equations by setting up global functions and operators from a simple element-local description. A general concept for incorporation of constraints eases the implementation of essential boundary conditions, hanging nodes and varying...

This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of $𝑎\mathrm{𝑝𝑟𝑖𝑜𝑟𝑖}-𝑎\mathrm{𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖}\mathrm{𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑒𝑠}$ is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.

We derive a posteriori error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature...