  Interpolation methods for nonlinear finite element discretizations are
commonly used to eliminate the computational costs associated with the repeated
assembly of the nonlinear systems. While the group finite element formulation
interpolates nonlinear terms onto the finite element approximation space, we
propose the use of a separate approximation space that is tailored to the
nonlinearity. In many cases, this allows for the exact reformulation of the
discrete nonlinear problem into a quadratic problem with algebraic constraints.
Furthermore, the substitution of the nonlinear terms often shifts general
nonlinear forms into trilinear forms, which can easily be described by
third-order tensors. The numerical benefits as well as the advantages in
comparison to the original group finite element method are studied using a wide
variety of academic benchmark problems.
