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.

翻译：近期提出的软有限元方法通过降低刚度（条件数）从而提升整体逼近精度。该方法在保持系统强制性的前提下，从有限元刚度双线性形式中减去一个最小二乘项，该惩罚项作用于网格界面上的梯度跳跃。本文提出软有限元方法的两种推广形式，旨在提升逼近精度并进一步降低离散系统的刚度。首先最自然的推广是：在质量双线性形式中添加最小二乘项。对于线性均匀单元，建立了特征值超收敛阶数为 $h^6$ 和 $h^8$ 的结果，其中 $h^8$ 是文献中已知的最高收敛阶数。其次，通过引入混合高斯型求积格式实现软有限元方法的推广。与传统有限元及软有限元方法相比，我们展示了刚度的进一步降低。强制性与最优误差收敛性分析沿用软有限元方法的研究框架，故本文聚焦于这些推广的数值研究。针对线性均匀单元，建立了解析特征对、精确特征值误差及超收敛误差分析体系。多种数值算例表明，广义软有限元方法在谱逼近领域（尤其是高频区域）具有显著潜力。