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Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation

Liliana Camargo, Sergio Rojas, Patrick Vega

TL;DR

This paper addresses efficient, accurate discretization of the Helmholtz problem at high wavenumbers using hybridizable discontinuous Galerkin (HDG) methods. It introduces two minimum-residual postprocessing schemes that generate superconvergent approximations of the scalar variable and supply built-in, locally computable a posteriori error indicators via local residual minimization on an enriched test space. The authors prove frequency-explicit a priori and a posteriori estimates, establish well-posedness of the local saddle-point postprocessing problems under mild mesh constraints, and validate the theory with 2D numerical experiments showing adaptive refinement driven by the indicators yields improved convergence and robustness against the pollution effect. The findings offer a scalable, parallelizable framework for adaptive HDG solvers in high-frequency wave propagation and lay groundwork for extensions to related time-harmonic problems such as Maxwell’s equations.

Abstract

We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, each yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted $H^1$-error for the scalar variable and the $L^2$-error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments.

Minimum-residual a posteriori error estimates for hybridizable discontinuous Galerkin discretizations of the Helmholtz equation

TL;DR

This paper addresses efficient, accurate discretization of the Helmholtz problem at high wavenumbers using hybridizable discontinuous Galerkin (HDG) methods. It introduces two minimum-residual postprocessing schemes that generate superconvergent approximations of the scalar variable and supply built-in, locally computable a posteriori error indicators via local residual minimization on an enriched test space. The authors prove frequency-explicit a priori and a posteriori estimates, establish well-posedness of the local saddle-point postprocessing problems under mild mesh constraints, and validate the theory with 2D numerical experiments showing adaptive refinement driven by the indicators yields improved convergence and robustness against the pollution effect. The findings offer a scalable, parallelizable framework for adaptive HDG solvers in high-frequency wave propagation and lay groundwork for extensions to related time-harmonic problems such as Maxwell’s equations.

Abstract

We propose and analyze two a posteriori error indicators for hybridizable discontinuous Galerkin (HDG) discretizations of the Helmholtz equation. These indicators are built to minimize the residual associated with a local superconvergent postprocessing scheme for the primal variable, measured in a dual norm of an enlarged discrete test space. The residual minimization is reformulated into equivalent local saddle-point problems, each yielding a superconvergent postprocessed approximation of the primal variable in the asymptotic regime for sufficiently regular exact solutions and a built-in residual representation with minimal computational effort. Both error indicators are based on frequency-dependent postprocessing schemes and verify reliability and efficiency estimates for a frequency-weighted -error for the scalar variable and the -error for the flux. We illustrate our theoretical findings through ad-hoc numerical experiments.
Paper Structure (15 sections, 12 theorems, 78 equations, 5 figures)

This paper contains 15 sections, 12 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Functional and discrete settings
  3. An HDG discretization of the Helmholtz equation
  4. The HDG discretization
  5. The HDG projection
  6. A priori error estimates
  7. Postprocessing schemes and a posteriori error indicators based on local residual minimization
  8. Well-posedness and a priori error estimates
  9. A posteriori error estimates
  10. Numerical experiments
  11. Plane wave solution
  12. Singular solution
  13. Conclusions
  14. Auxiliary results
  15. A priori error estimates for the postprocessed approximation based on the Helmholtz operator.

Key Result

Theorem 3.1

If $(\boldsymbol v,w):K\to\mathbb{C}^d\times\mathbb{C}$ is sufficiently smooth and $\tau\in\mathcal{R}_0(\partial K)$, then system eq:proj is uniquely solvable for $\Pi_{\boldsymbol V}\boldsymbol v$ and $\Pi_W w$. Furthermore, there is a constant $C$ independent of $K$, $\omega$ and $\tau$ such that for $l_w,l_{\boldsymbol v}\in[0,k]$, $k\in\mathbb{N}$, where $\tau_K^{\max}:=\max\tau|_{\partial K}

Figures (5)

  • Figure 1: Increment in the convergence rates of the $L^2$-error, uniform refinement, plane wave solution.
  • Figure 2: $\mathrm{e}_h^\pm$ and $\eta_h^\pm$ vs. Nel$^{1/2}$ , uniform refinement, plane wave solution.
  • Figure 3: Effectivity index $\xi_h^\pm$ vs. Nel, uniform refinement, plane wave solution.
  • Figure 4: $\omega=\pi$, $\nu_h^-$, and $\eta_h^-$; adaptive refinement, singular solution.
  • Figure 5: $\omega=16\sqrt{2}\pi$, $\nu_h^\pm$, and $\eta_h^\pm$; adaptive refinement, singular solution.

Theorems & Definitions (24)

  • Theorem 3.1: MR3970243
  • Theorem 3.2: A priori error estimates for HDG
  • Theorem 4.1: A priori error estimates for the postprocessed approximations
  • Theorem 4.3: Reliability
  • Theorem 4.4: Efficiency
  • Remark 4.5
  • Theorem 4.6: Well-posedness and stability estimates
  • proof
  • Lemma 4.7
  • proof : Proof of Lemma \ref{['lemma:apriori_secondpostpm']} for $\nu_h^-$
  • ...and 14 more