  The recently proposed soft finite element method (SoftFEM) reduces the
stiffness (condition numbers), consequently improving the overall approximation
accuracy. The method subtracts a least-square term that penalizes the gradient
jumps across mesh interfaces from the FEM stiffness bilinear form while
maintaining the system's coercivity. Herein, we present two generalizations for
SoftFEM that aim to improve the approximation accuracy and further reduce the
discrete systems' stiffness. Firstly and most naturally, we generalize SoftFEM
by adding a least-square term to the mass bilinear form. Superconvergent
results of rates $h^6$ and $h^8$ for eigenvalues are established for linear
uniform elements; $h^8$ is the highest order of convergence known in the
literature. Secondly, we generalize SoftFEM by applying the blended
Gaussian-type quadratures. We demonstrate further reductions in stiffness
compared to traditional FEM and SoftFEM. The coercivity and analysis of the
optimal error convergences follow the work of SoftFEM. Thus, this paper focuses
on the numerical study of these generalizations. For linear and uniform
elements, analytical eigenpairs, exact eigenvalue errors, and superconvergent
error analysis are established. Various numerical examples demonstrate the
potential of generalized SoftFEMs for spectral approximation, particularly in
high-frequency regimes.
