Favored by the immense progress in information technology, numerical simulations based on the finite element method have become an indispensable tool in engineering science as well as industry. The quality and thus the validity of simulation results strongly depends on the underlying mathematical models. On the one hand, this concerns the employed material models, which are constantly being further developed in order to be able to represent even the most complex mechanical effects and phenomena. On the other hand, the choice of the finite element formulation plays a decisive role. Due to their simplicity, robustness and efficiency, element formulations based on low-order shape functions are preferred. However, it is known that such formulations behave much too stiffly, for example, in the case of bending dominated deformations as well as in the case of nearly incompressible materials, thus distorting the quality of the simulation results. Therefore, the main goal of the research field of finite element technology is to address these stiffening (locking) effects of low polynomial order elements in order to allow the most versatile use of the respective formulation. This cumulative dissertation aims to make a valuable contribution in this regard. It consists essentially of three peer-reviewed journal articles in the subject area already published by the author (and his co-authors). Herein, the main goal is to extend a class of low-order continuum elements, which are based on the technology of reduced integration with hourglass stabilization, to the analysis of gradient-extended damage. In this context, gradient-extended damage is used to address the pathological mesh dependence in finite element simulations in the presence of softening material behavior.