This paper presents a new augmented finite element method (A-EEM) that can account for path-arbitrary, multiple intraelemental discontinuities with a demonstrated improvement in numerical efficiency by two orders of magnitude when compared to the extended finite element method {X-EEM). We show that the new formulation enables the derivation of explicit, fully condensed elemental equilibrium equations that are mathematically exact within the finite element context. More importantly, it allows for repeated elemental augmentation to include multiple interactive cracks within a single element without additional external nodes or degrees of freedom (DoEs). A novel algorithm that can rapidly and accurately solve the nonlinear equilibrium equations at the elemental level has also been developed for cohesive cracks with piecewise linear traction-separation laws. This efficient new solving algorithm, coupled with the mathematically exact elemental equilibrium equation, leads to dramatic improvement in numerical accuracy, efficiency, and stability when dealing with arbitrary cracking problems. The A-EEM's excellent capability in high-fidelity simulation of interactive cohesive cracks in homogeneous and heterogeneous solids has been demonstrated through several numerical examples.