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The numerical linear algebra of weights: from the spectral analysis to conditioning and preconditioning in the Laplacian case

Ludovico Bruni Bruno, Matteo Semplice, Stefano Serra-Capizzano

TL;DR

The paper develops a spectral framework for weight-based finite element methods in one dimension, using FEEC and GLT/Toeplitz theory to connect local weight placements to global spectral properties of the stiffness matrix. It shows that the stiffness sequence is described by a matrix-valued symbol f_r^ξ(θ) whose eigenvalues separate, leading to a conditioning growth of order n^2; by optimizing the weight locations ξ (notably near Gauss–Lobatto-like points), conditioning improves and efficient preconditioners (circulant-based with Strang corrections) can be constructed. The approach yields robust preconditioning for constant and variable coefficients, extends to nonuniform and graded meshes, and demonstrates favorable convergence and stability in numerical experiments. The work points to feasible extensions to higher dimensions and other FEEC elements, with GLT theory providing a principled route to analyze conditioning and preconditioners in more complex settings.

Abstract

Weights are geometrical degrees of freedom that allow to generalise Lagrangian finite elements. They are defined through integrals over specific supports, well understood in terms of differential forms and integration, and lie within the framework of finite element exterior calculus. In this work we exploit this formalism with the target of identifying supports that are appealing for finite element approximation. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. We describe the conditioning and the spectral global behavior in terms of the standard Toeplitz machinery and GLT theory, leading to the identification of the optimal choices for weights. Moreover, we propose and test ad hoc preconditioners, in dependence of the discretization parameters and in connection with conjugate gradient method. The model problem we consider is a onedimensional Laplacian, both with constant and non constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, demonstrating the advantages of weights-induced bases over standard Lagrangian ones. Open problems and future steps are listed in the conclusive section, especially regarding the multidimensional case.

The numerical linear algebra of weights: from the spectral analysis to conditioning and preconditioning in the Laplacian case

TL;DR

The paper develops a spectral framework for weight-based finite element methods in one dimension, using FEEC and GLT/Toeplitz theory to connect local weight placements to global spectral properties of the stiffness matrix. It shows that the stiffness sequence is described by a matrix-valued symbol f_r^ξ(θ) whose eigenvalues separate, leading to a conditioning growth of order n^2; by optimizing the weight locations ξ (notably near Gauss–Lobatto-like points), conditioning improves and efficient preconditioners (circulant-based with Strang corrections) can be constructed. The approach yields robust preconditioning for constant and variable coefficients, extends to nonuniform and graded meshes, and demonstrates favorable convergence and stability in numerical experiments. The work points to feasible extensions to higher dimensions and other FEEC elements, with GLT theory providing a principled route to analyze conditioning and preconditioners in more complex settings.

Abstract

Weights are geometrical degrees of freedom that allow to generalise Lagrangian finite elements. They are defined through integrals over specific supports, well understood in terms of differential forms and integration, and lie within the framework of finite element exterior calculus. In this work we exploit this formalism with the target of identifying supports that are appealing for finite element approximation. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. We describe the conditioning and the spectral global behavior in terms of the standard Toeplitz machinery and GLT theory, leading to the identification of the optimal choices for weights. Moreover, we propose and test ad hoc preconditioners, in dependence of the discretization parameters and in connection with conjugate gradient method. The model problem we consider is a onedimensional Laplacian, both with constant and non constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, demonstrating the advantages of weights-induced bases over standard Lagrangian ones. Open problems and future steps are listed in the conclusive section, especially regarding the multidimensional case.
Paper Structure (23 sections, 10 theorems, 79 equations, 10 figures, 6 tables)

This paper contains 23 sections, 10 theorems, 79 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Weights-based FEM
  3. Bases in duality
  4. Weights
  5. Bilinear form and stiffness matrix
  6. Spectral tools and useful matrix structures
  7. Clustering and distribution
  8. Block Toeplitz/Circulant/diagonal matrices and zero-distributed matrix-sequences
  9. The $*$-algebra of block GLT sequences
  10. Spectral symbol and spectral theory
  11. Determinant of the symbol
  12. Extremal eigenvalues and conditioning
  13. Optimization of weights
  14. Constant coefficients
  15. Non-constant coefficients
  16. ...and 8 more sections

Key Result

Proposition 2.2

Let $A$ and $B$ be the stiffness matrices written with respect to the bases $\{\varphi_i\}_{i=1}^{N}$ and respectively $\{\omega_i\}_{i=1}^{N}$; there exists an invertible matrix $M$ such that Moreover, the elemental matrices $A^{(T_k)}$ and $B^{(T_k)}$ for $T_k\in\mathcal{T}$ written with respect to $\{\widehat{\varphi}_i\}_{i=1}^{N_{\text{\tiny{loc}}}}$ and respectively to $\{\widehat{\omega}_i

Figures (10)

  • Figure 1: Structure of the stiffness matrix and of the Toeplitz matrix. Element contributions are hatched in black: note the overlap of nearby blocks and the smaller first and last block. The blocks of the Toeplitx matrix are shaded in colour.
  • Figure 2: Eigenvalues functions of $f_3^\xi (\theta)$ for $\xi = 0.28$ (left) and $\xi= 0.10$ (right).
  • Figure 3: Eigenvalues functions of $f_4^\xi (\theta)$ for $\xi = 0.12$ (left) and $\xi = 0.22$ (right).
  • Figure 4: A plot of the conditioning $\kappa_2(n, \xi)$ as $\xi$ varies, in logarithmic scale. Here the diameter of the interval is $h = 1/64$ so that the matrix size $n$ is fixed and the red line represents the conditioning with respect to uniform nodes, namely those in duality with usual Lagrangian elements. The cases $r = 3$ (left) and $r = 4$ (right) are depicted.
  • Figure 5: A plot of the spectrum of the Laplacian with constant coefficients. Solid line represents the continuous spectrum, dashed lines represent the symbol associated with different mesh sizes. Lagrange equispaced elements (left) and optimized weights (right) are compared. The degree $r = 3$ is depicted.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.5
  • Lemma 4.1
  • ...and 17 more