We present novel overlapping finite elements for a new paradigm of solution proposed in our previous papers, see Bathe (2016) and Bathe and Zhang (2017).We give the formulation of the new overlapping elements and the solutions of basic numerical examples to investigate the robustness and efficiency of the new finite elements. The results show that the new overlapping elements are quite distortion insensitive and the numerical integration of the element matrices is efficient. The computational effort to integrate the matrices is much less than in meshfree methods. Finally, we illustrate the complete solution scheme of the new paradigm using the overlapping finite elements in the analysis of the bracket problem already considered in Bathe and Zhang (2017).While the paper proposes and studies new overlapping elements, we conclude that further research is needed to fully harvest the potential of the new analysis approach.