When modeling shearing or cutting processes, as found in geomechanics or manufacturing, the simulated bodies often undergo locally extreme deformations. These applications are often modeled using particle methods to avoid mesh distortion and entanglement arising in mesh-based techniques, such as the finite element method. However, they also incur additional computational effort. The combination of a mesh discretization with a particle discretization aims to optimize these simulations by employing meshes in areas with small and medium deformation and particles in areas with extreme deformations. In this contribution, the finite element method is coupled with the material point method to create a robust and efficient framework for simulations with partially extreme deformations. An implicit formulation of the methods is used to enable stable and accurate simulations of transient and quasi-static processes. To enable an accurate, parameter-free and mesh-independent coupling, the dual-mortar-approach is used, exploiting dual shape functions to efficiently resolve the coupling constraints. Furthermore, an automatic conversion scheme is proposed, adaptively converting finite elements to material points upon distortion.