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Spectral analysis of the stiffness matrix sequence in the approximated Stokes equation

Samuele Ferri, Chiara Giraudo, Valerio Loi, Miroslav Kuchta, Stefano Serra-Capizzano

TL;DR

This work analyzes the spectral properties of the stiffness and divergence matrices arising from a Taylor–Hood $P_2$–$P_1$ discretization of the 2D Stokes problem with variable viscosity. By leveraging Generalized Locally Toeplitz (GLT) theory, the authors derive asymptotic eigenvalue and singular value distributions for the block matrices $A_n$ and $B_n$, representing the diffusion and divergence operators, respectively, and express these distributions via matrix-valued symbols that factor through viscosity, i.e. $f^{[m]}(x,y, heta_1, heta_2)= u^{[m]}(x,y) ilde{G}( heta_1, heta_2)$. Through carefully designed permutations, projections, and compressions, the block structures are embedded into a GLT framework, yielding a precise spectral description and guiding the construction of a preliminary GLT-based preconditioner. Numerical tests across constant and variable viscosity inputs confirm adherence to the predicted symbols, demonstrate absence of spectral outliers, and show improved convergence behavior with the proposed preconditioner. The results offer a principled way to analyze and precondition variable-viscosity Stokes systems and point to practical extensions to more general discretizations and applications such as geophysics and glacial isostatic adjustment.

Abstract

In the present paper, we analyze in detail the spectral features of the matrix sequences arising from the Taylor-Hood $\mathbb{P}_2$-$\mathbb{P}_1$ approximation of variable viscosity for $2d$ Stokes problem under weak assumptions on the regularity of the diffusion. Localization and distributional spectral results are provided, accompanied by numerical tests and visualizations. A preliminary study of the impact of our findings on the preconditioning problem is also presented. A final section with concluding remarks and open problems ends the current work.

Spectral analysis of the stiffness matrix sequence in the approximated Stokes equation

TL;DR

This work analyzes the spectral properties of the stiffness and divergence matrices arising from a Taylor–Hood – discretization of the 2D Stokes problem with variable viscosity. By leveraging Generalized Locally Toeplitz (GLT) theory, the authors derive asymptotic eigenvalue and singular value distributions for the block matrices and , representing the diffusion and divergence operators, respectively, and express these distributions via matrix-valued symbols that factor through viscosity, i.e. . Through carefully designed permutations, projections, and compressions, the block structures are embedded into a GLT framework, yielding a precise spectral description and guiding the construction of a preliminary GLT-based preconditioner. Numerical tests across constant and variable viscosity inputs confirm adherence to the predicted symbols, demonstrate absence of spectral outliers, and show improved convergence behavior with the proposed preconditioner. The results offer a principled way to analyze and precondition variable-viscosity Stokes systems and point to practical extensions to more general discretizations and applications such as geophysics and glacial isostatic adjustment.

Abstract

In the present paper, we analyze in detail the spectral features of the matrix sequences arising from the Taylor-Hood - approximation of variable viscosity for Stokes problem under weak assumptions on the regularity of the diffusion. Localization and distributional spectral results are provided, accompanied by numerical tests and visualizations. A preliminary study of the impact of our findings on the preconditioning problem is also presented. A final section with concluding remarks and open problems ends the current work.

Paper Structure

This paper contains 34 sections, 7 theorems, 85 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology and notation
  3. Spectral analysis tools
  4. Asymptotic spectral symbol and approximation tools
  5. Toeplitz structures and GLT tools
  6. Definition and properties of the block multilevel GLT class
  7. Block-structured sequences and the extradimensional approach
  8. The block multilevel structures
  9. The block multilevel structure of A_n
  10. Simplifying the block structure by using projections
  11. Diagonal blocks Di.
  12. Off-diagonal blocks B_i.
  13. Blocks C_i.
  14. Spectral analysis of A_n
  15. Toeplitz symbol
  16. ...and 19 more sections

Key Result

Theorem 3.4

Let $\{A_n\}_{n}$ and $\{B_{n,m}\}_{n}$ be sequences of matrices, and let $f, f_{m}: D \subset \mathbb{R}^{k} \rightarrow$$\mathbb{C}$ be measurable functions. Suppose the following conditions hold: Then it follows that $\{A_n\}_{n} \sim_{\sigma} f$. If all the sequences are Hermitian, the distributional results also hold for the eigenvalue symbol.

Figures (6)

  • Figure 1: Effect of variable viscosity \ref{['eq:viscosity']} on conditioning of the Stokes model \ref{['eq: Stokes equations']} using the preconditioner \ref{['eq:wathen']}.
  • Figure 2: Ordering for the cells/viscosity (left), $\mathbb{P}_2$ degrees of freedom for velocity components (center) and $\mathbb{P}_1$ pressures (right) for case $n=4$. Note that only the interior velocity degrees of freedom, i.e. the basis functions in $(H^1_0(\Omega))^2$, enter into the linear system.
  • Figure 3: Adherence of the eigenvalues of $A_{x,n}$ to its symbol for $n=16$. The first line displays the plots corresponding to Group 1 (on the left) and Group 2 (on the right), while the second line shows Group 3; from left $\gamma = 1$, $\gamma = 10$, $\gamma = 100$.
  • Figure 4: Adherence of the singular values to its symbol for $n=16$. The plot on the left corresponds to the matrix sequence $B_{x,n}$ and the one on the right to $B_{y,n}$.
  • Figure 5: Adherence of the eigenvalues of $M_n$ to its symbol for $n=16$. The first line displays the plots corresponding to Group 1 (on the left) and Group 2 (on the right), while the second line shows Group 3; from left $\gamma = 1$, $\gamma = 10$, $\gamma = 100$.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Example 1: Variable viscosity and standard Stokes preconditioner
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.2: rect-glt
  • Theorem 3.4: GS-I
  • Theorem 3.5
  • Theorem 3.6: GS-I
  • Definition 3.3: Toeplitz, block-Toeplitz, block multilevel Toeplitz matrices
  • ...and 6 more