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Generalised Soft Finite Element Method for Elliptic Eigenvalue Problems

Jipei Chen, Victor M. Calo, Quanling Deng

TL;DR

This work addresses the stiffness and high-frequency accuracy limitations of standard Galerkin FEM for elliptic eigenvalue problems by generalizing SoftFEM. It introduces two generalizations: adding a gradient-jump penalty to the mass (GSFEM) and applying blending quadratures (SoftFEMBQ), with their combination (GSFEMBQ); these yield new generalized eigenproblems that reduce stiffness while preserving coercivity. In 1D, analytical results show superconvergence in eigenvalues up to order $O(h^8)$ for certain parameter choices and quantify asymptotic stiffness reductions (e.g., $\rho_{gsq,\infty}=257/160$ for a particular setting). Numerical experiments in 1D and 2D confirm substantial stiffness reductions and improved high-frequency spectral accuracy, including problems with variable diffusion $\kappa$, highlighting enhanced robustness of the generalized SoftFEM methods for spectral approximation. Overall, the proposed GSFEM families offer a practical route to lower condition numbers and more accurate spectra in high-frequency regimes, with potential extensions to higher-order elements and soft isogeometric analysis.

Abstract

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.

Generalised Soft Finite Element Method for Elliptic Eigenvalue Problems

TL;DR

This work addresses the stiffness and high-frequency accuracy limitations of standard Galerkin FEM for elliptic eigenvalue problems by generalizing SoftFEM. It introduces two generalizations: adding a gradient-jump penalty to the mass (GSFEM) and applying blending quadratures (SoftFEMBQ), with their combination (GSFEMBQ); these yield new generalized eigenproblems that reduce stiffness while preserving coercivity. In 1D, analytical results show superconvergence in eigenvalues up to order for certain parameter choices and quantify asymptotic stiffness reductions (e.g., for a particular setting). Numerical experiments in 1D and 2D confirm substantial stiffness reductions and improved high-frequency spectral accuracy, including problems with variable diffusion , highlighting enhanced robustness of the generalized SoftFEM methods for spectral approximation. Overall, the proposed GSFEM families offer a practical route to lower condition numbers and more accurate spectra in high-frequency regimes, with potential extensions to higher-order elements and soft isogeometric analysis.

Abstract

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 and for eigenvalues are established for linear uniform elements; 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.
Paper Structure (19 sections, 6 theorems, 49 equations, 7 figures, 8 tables)

This paper contains 19 sections, 6 theorems, 49 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Problem Statement and SoftFEM
  3. Problem Statement
  4. Galerkin FEM
  5. SoftFEM
  6. Generalized SoftFEM
  7. Generalization by Adding Gradient-Jump Penalty in Mass
  8. Generalization by Blending Quadratures
  9. Analytical Results of Linear Elements 1D
  10. GSFEM
  11. SoftFEMBQ
  12. Combination of GSFEM and SoftFEMBQ
  13. Numerical Experiments
  14. Study on Superconvergence
  15. Study on Stiffness Reduction
  16. ...and 4 more sections

Key Result

Lemma 4.1

For GSFEM approximation with $N^h$ uniform elements, the GMEVP $\mathbf{K}_{s} \mathbf{U}_{gs} = \lambda^h_{gs} \mathbf{M}_{gs} \mathbf{U}_{gs}$ in eq:GeneSoftFEM has eigenpairs $(\lambda_{gs,j}^h, \mathbf{U}_{gs,j})$ for all $j\in\{1,\cdots,N^h-1\}$ with where $t_j:=j \pi h$, $\mathbf{U}_{gs,j,k}$ is the $k$-th component of the $j$-th eigenvector $\mathbf{U}_{gs,j}$, $c_j> 0$ is some normalizati

Figures (7)

  • Figure 1: Comparison of Galerkin FEM (black), SoftFEM (blue), and GSFEM (red) for the spectral approximation of the Laplacian eigenvalue problem in 1D using $N^h=100$ elements with polynomial degrees $p\in \{1, 2, 3\}$ (from top to bottom row).
  • Figure 2: Eigenvalue superconvergence (red lines) for various methods with linear uniform elements with size $N^{h}=32$. The black dotted lines are reference lines with theoretic orders (see Section \ref{['sec:ana']}) 6, 6, 8, and 6, respectively, for GSFEM (top left), SoftFEMBQ (top right), GSFEMBQ with $\eta_K = \frac{31}{252}, \eta_M = \frac{23}{3780}$ and $\alpha = \frac{26}{21}$ (bottom left), and GSFEMBQ with $\eta_K = -\frac{1}{12}, \eta_M = -\frac{1}{90}$ and $\alpha = 0$ (bottom right).
  • Figure 3: Condition number $\sigma_{gs}$ of 1D Laplace eigenvalue problem when using GSFEM with $N^{h}=200$ uniform elements with $p=1$ (left), $p=2$ (middle), and $p=3$ (right).
  • Figure 4: Relative eigenvalue errors for the 2D Laplace eigenvalue problem when using quadratic Galerkin FEM, SoftFEM and GSFEM with a mesh of size $40\times 40$.
  • Figure 6: Comparison of eigenvalue errors (left) and eigenfunction errors (right) using different methods for Laplacian eigenvalue problem in 1D with $N^h=200$ uniform elements: Galerkin FEM (black), SoftFEM (blue), SoftFEMBQ (green), and GSFEMBQ (magenta). The parameters are chosen as in Table \ref{['tab:result1']} and the blending parameter $\alpha=0.95$.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Remark 3.1: GSFEM
  • Remark 3.2: SoftFEMBQ
  • Remark 3.3: Alternative Combination
  • Lemma 4.1: Analytical eigenvalues and eigenvectors
  • proof
  • Remark 4.2
  • Theorem 4.3: Eigenvalue superconvergence
  • proof
  • Remark 4.4
  • Remark 4.5
  • ...and 10 more