We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a least-squares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts. The penalty term is designed to dampen the high frequencies in the spectrum and so it is weighted here by a negative coefficient. The resulting approximation technique is called softFEM since it reduces the stiffness of the problem. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. We derive a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form. Then we prove that softFEM delivers the same optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We next compare the discrete eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a sensible choice for the softness parameter. With this choice, the stiffness reduction ratio scales linearly with the polynomial degree. Various numerical experiments illustrate the benefits of using softFEM over Galerkin FEM.

我们提出、数学分析和数值研究了一种用于二阶椭圆算子谱的有限元近似的新方法。主要思想是通过从标准刚度双线性形式中减去跨网格界面的梯度跳跃的最小二乘惩罚来降低问题的刚度。这种惩罚双线性形式类似于用于在各种上下文中稳定有限元近似的已知技术。惩罚项旨在抑制频谱中的高频，因此在此处由负系数加权。由此产生的近似技术称为 softFEM，因为它降低了问题的刚度。与标准 Galerkin FEM 相比，softFEM 的两个主要优点是改进了离散谱上部特征值的近似值和减少了刚度矩阵的条件数。我们推导出对稳定双线性形式加权的软度参数的尖锐上限，以保持 softFEM 双线性形式的矫顽力。然后我们证明 softFEM 提供与特征值和特征向量的标准 Galerkin FEM 近似相同的最佳收敛率。我们接下来比较使用 Galerkin FEM 和 softFEM 时获得的离散特征值。最后，对一维拉普拉斯特征值问题的线性 softFEM 的详细分析为软度参数提供了一个明智的选择。通过这种选择，刚度降低率与多项式次数成线性关系。