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Stability of conforming space-time isogeometric methods for the wave equation

Matteo Ferrari, Sara Fraschini

Abstract

We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2018), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2023), have introduced unconditionally stable schemes by adding non-consistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results.

Stability of conforming space-time isogeometric methods for the wave equation

Abstract

We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2018), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2023), have introduced unconditionally stable schemes by adding non-consistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results.
Paper Structure (20 sections, 15 theorems, 127 equations, 9 figures, 1 table)

This paper contains 20 sections, 15 theorems, 127 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model problem and stabilized variational formulations
  3. Maximal regularity splines and associated matrices
  4. Conditioning of Toeplitz band matrices
  5. Conditioning of Toeplitz band matrices with symmetries
  6. Conditioning of matrices defined by maximal regularity splines
  7. Necessary condition for the weakly well-conditioning of $\{\mathbf{K}_n^p(\rho,\delta)\}_n$
  8. Sufficient condition for the weakly well-conditioning of $\{\mathbf{K}_n^p(\rho,\delta)\}_n$
  9. Alternative IgA stabilizations
  10. Numerical results
  11. Conclusion
  12. The proof of Theorem \ref{['th:45']}
  13. The last assumption of Theorem \ref{['th:45']} for $\{\mathbf{K}_n^p(\rho,\delta)\}_n$
  14. The case $p=2$
  15. The case $p=3$
  16. ...and 5 more sections

Key Result

Proposition 3.2

For all $p \ge 1$, let $\mathbf{M}_h^p, \mathbf{B}_h^p$ and $\mathbf{D}_h^p$ be defined in eq:3.2. Then, the following properties hold true.

Figures (9)

  • Figure 1: The spline basis of $S_{h}^p(0,1)$ with $N = 5$. The break points are indicated by vertical dashed lines, and central splines are highlighted with thicker lines.
  • Figure 2: For various $k \equiv k(p)$, by varying $p$, the values of $|\delta_p^k|$, as in \ref{['eq:63']}, on a semi-logarithmic scale.
  • Figure 3: By varying $M$, the sequence $\{C_M\}_M$, as in \ref{['eq:64']}, on a semi-logarithmic scale.
  • Figure 4: Spectral condition numbers $\kappa_2(\mathbf{K}^p_n(\rho,0))$ in semi-logarithmic scale, with $n=1000$ by varying $\rho \in [8,13]$, with $p \in \{1,2,3\}$.
  • Figure 5: Spectral condition numbers $\kappa_2(\mathbf{K}^p_n(\rho,0))$ in semi-logarithmic scale, with $n=2000$ by varying $\rho \in [9.84,9.92]$, with $p \in \{4,5\}$.
  • ...and 4 more figures

Theorems & Definitions (47)

  • Example 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.5
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • ...and 37 more