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Hybrid discontinuous Galerkin discretizations for the damped time-harmonic Galbrun's equation

Martin Halla, Christoph Lehrenfeld, Tim van Beeck

TL;DR

The paper tackles the forward problem for the damped time-harmonic Galbrun equation in solar and stellar oscillations. It develops hybrid discontinuous Galerkin discretizations that are stable and optimally convergent for all $k \ge 1$, with robustness to large parameter contrasts, by exploiting a weak T-coercivity framework and casting the discrete scheme as a discrete approximation scheme (DAS). A constructive Helmholtz-type decomposition yields discrete operators $T_n$ and a regularity theory that guarantees stability and convergence, including explicit convergence rates under mild Mach-number assumptions. Numerical experiments validate the theory, demonstrate Mach-number robustness, compare with SIP, and show practical efficiency gains through static condensation and lifting stabilization, making the approach suitable for astrophysical simulations. The work provides a rigorous, scalable HDG framework for accurate and efficient helioseismology computations with complex, heterogeneous coefficients.

Abstract

In this article, we study the damped time-harmonic Galbrun's equation which models solar and stellar oscillations. We introduce and analyze hybrid discontinuous Galerkin discretizations (HDG) that are stable and optimally convergent for all polynomial degrees greater than or equal to one. The proposed methods are robust with respect to the drastic changes in the magnitude of the coefficients that naturally occur in stars. Our analysis is based on the concept of discrete approximation schemes and weak T-compatibility, which exploits the weakly T-coercive structure of the equation. Compared to the $H^1$-conforming discretization of [Halla, Lehrenfeld, Stocker, 2022], our method offers improved stability and robustness. Furthermore, it significantly reduces the computational costs compared to the $H(\operatorname{div})$-conforming DG discretization of [Halla, 2023], which has similar stability properties. These advantages make the proposed HDG methods well-suited for astrophysical simulations.

Hybrid discontinuous Galerkin discretizations for the damped time-harmonic Galbrun's equation

TL;DR

The paper tackles the forward problem for the damped time-harmonic Galbrun equation in solar and stellar oscillations. It develops hybrid discontinuous Galerkin discretizations that are stable and optimally convergent for all , with robustness to large parameter contrasts, by exploiting a weak T-coercivity framework and casting the discrete scheme as a discrete approximation scheme (DAS). A constructive Helmholtz-type decomposition yields discrete operators and a regularity theory that guarantees stability and convergence, including explicit convergence rates under mild Mach-number assumptions. Numerical experiments validate the theory, demonstrate Mach-number robustness, compare with SIP, and show practical efficiency gains through static condensation and lifting stabilization, making the approach suitable for astrophysical simulations. The work provides a rigorous, scalable HDG framework for accurate and efficient helioseismology computations with complex, heterogeneous coefficients.

Abstract

In this article, we study the damped time-harmonic Galbrun's equation which models solar and stellar oscillations. We introduce and analyze hybrid discontinuous Galerkin discretizations (HDG) that are stable and optimally convergent for all polynomial degrees greater than or equal to one. The proposed methods are robust with respect to the drastic changes in the magnitude of the coefficients that naturally occur in stars. Our analysis is based on the concept of discrete approximation schemes and weak T-compatibility, which exploits the weakly T-coercive structure of the equation. Compared to the -conforming discretization of [Halla, Lehrenfeld, Stocker, 2022], our method offers improved stability and robustness. Furthermore, it significantly reduces the computational costs compared to the -conforming DG discretization of [Halla, 2023], which has similar stability properties. These advantages make the proposed HDG methods well-suited for astrophysical simulations.

Paper Structure

This paper contains 26 sections, 19 theorems, 159 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Abstract framework
  3. Weak T-coercivity
  4. Discrete approximation schemes and weak T-compatibility
  5. Preliminaries & continuous formulation
  6. Preliminaries
  7. Continuous weak formulation
  8. Hybrid discontinuous Galerkin discretizations
  9. Discrete spaces
  10. Discrete weak formulation & stabilization mechanisms
  11. Projection operators
  12. Projection operators
  13. Interpretation as DAS
  14. Convergence Analysis
  15. Construction of $T$ and $T_n$
  16. ...and 11 more sections

Key Result

Lemma 3

[lemma]lem:RegularStable Let $A \in L(X)$ be bijective and $(X_n,p_n, A_n)$ be a DAS of $(X,A)$. If $(A_n)_{n \in \mathbb{N}}$ is regular, then $(A_n)_{n \in \mathbb{N}}$ is stable. Further, if $u \in X$ solves $Au=f$ and $u_n \in X_n$ are solutions to $A_n u_n = f_n$ where $\lim_{n \rightarrow \inf

Figures (9)

  • Figure 1: We compare the coupling of unknowns for DG and HDG methods. For illustration purposes, we depict the situation in the scalar setting with polynomial degree $k = 3$.
  • Figure 2: Roadmap for the analysis of the discrete problem \ref{['eq:discr:weakForm']}.
  • Figure 3: We compare the sparsity pattern of the stiffness matrix obtained with the following four methods: $H(\operatorname{div})$-conforming DG with lifting stabilization (left), $H(\operatorname{div})$-conforming DG with SIP (middle-left), $H(\operatorname{div})$-conforming HDG with lifting stabilization (middle-right), and $H(\operatorname{div})$-conforming HDG with SIP (right). For the HDG methods, we use static condensation and for the DG method with the lifting operator, we apply the Schur complement to eliminate the unknowns associated with the lifting operator. In the HDG setting, both methods lead to the same number of non-zero entries ($\texttt{nzes}$) (even though the couplings differ slightly), whereas in the DG setting, the lifting operator almost doubles the number of non-zero entries. For the computations, we chose a mesh with $27$ elements and the polynomial degree $k = 5$.
  • Figure 4: Convergence of the methods listed in \ref{['table:discretizations']} against \ref{['eq:num:refsol']} for polynomial degrees $k = 3$ and $k = 4$ with Mach number $\Vert c_s^{-1} \mathbf{b}_{c_s} \Vert^2_{\mathbf{L}^\infty} = 0.25$. For the fully non-conforming methods, we consider the choices $\alpha \in \{10 k^2,100 k^2\}$ (dashed, solid). The error is measured in the $\mathbb{X}(\mathcal{T}_n)$-norm. In the embedded bar charts we show the number of coupled degrees of freedom for each method at the last refinement level $L=4$.
  • Figure 5: $\mathbb{X}(\mathcal{T}_n)$-error of the ${\mathbf H}^1$-conforming discretization, the $H(\operatorname{div})$-conforming HDG discretization and the respective best-approximation error for the flows $\mathbf{b}_{\eta}$, $\eta \in \{1,c_s,c_s/\rho\}$ modeled after \ref{['eq:numex:BFlows']}. We use the coefficient $c_{\mathbf{b}}$ to vary the Mach number in between $0.05$ and $1.25$.
  • ...and 4 more figures

Theorems & Definitions (46)

  • Definition 1: Weak T-coercivity
  • Definition 2
  • Lemma 3: Lem. 1 & 2 of HLS22H1
  • Theorem 4: Thm. 3 of HLS22H1
  • Remark 5: Hybridization as the enabler of static condensation
  • Remark 6: Alternative choices for $\mathbb{X}_{\mathcal{T}_n}$ and $\mathbb{X}_{\mathcal{F}_n}$
  • Remark 7: Stabilization in the setting of \ref{['rem:HdivHDG']}
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 36 more