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Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

T. Chaumont-Frelet

TL;DR

This work analyzes interior penalty discontinuous Galerkin discretizations for the Helmholtz equation under minimal regularity, developing a duality-based Aubin–Nitsche framework to obtain both a priori and a posteriori error estimates. A lifting-based duality argument handles nonconformity without requiring high solution regularity, yielding quasi-optimal error control in the broken energy norm $|||u-u_h|||_{oldsymbol\omega}$ and computable residual-based estimators. The paper introduces broken-norm approximation factors that quantify dual-approximation efficiency and proves reliability and efficiency results for a residual-based estimator, enabling robust hp-adaptive strategies at high frequency. Overall, the results bridge conforming and IPDG analyses for wave problems, providing rigorous error control tools for adaptive Helmholtz simulations with minimal regularity assumptions and nonconforming discretizations.

Abstract

We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.

Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

TL;DR

This work analyzes interior penalty discontinuous Galerkin discretizations for the Helmholtz equation under minimal regularity, developing a duality-based Aubin–Nitsche framework to obtain both a priori and a posteriori error estimates. A lifting-based duality argument handles nonconformity without requiring high solution regularity, yielding quasi-optimal error control in the broken energy norm and computable residual-based estimators. The paper introduces broken-norm approximation factors that quantify dual-approximation efficiency and proves reliability and efficiency results for a residual-based estimator, enabling robust hp-adaptive strategies at high frequency. Overall, the results bridge conforming and IPDG analyses for wave problems, providing rigorous error control tools for adaptive Helmholtz simulations with minimal regularity assumptions and nonconforming discretizations.

Abstract

We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.
Paper Structure (21 sections, 11 theorems, 113 equations)

This paper contains 21 sections, 11 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Setting and preliminary results
  3. Setting
  4. Functional spaces
  5. Helmholtz problem
  6. Computational mesh
  7. Polynomial spaces
  8. Broken Sobolev spaces
  9. Jumps, averages, lifting and discrete gradient
  10. Broken norms
  11. IPDG form
  12. The discrete Helmholtz problem
  13. Conforming subspaces and projections
  14. Approximation factors
  15. Duality under minimal regularity
  16. ...and 6 more sections

Key Result

Lemma 3.1

For all $\phi \in H^1(\mathcal{T}_h)$ and $\boldsymbol \sigma \in \boldsymbol X_{\Gamma_{\rm N}}(\operatorname{div},\Omega)$, we have and in particular for all $\widetilde{\phi} \in H^1_{\Gamma_{\rm D}}(\Omega)$ and $\boldsymbol \sigma_h \in \boldsymbol{\mathcal{P}}_p(\mathcal{T}_h) \cap \boldsymbol X_{\Gamma_{\rm N}}(\operatorname{div},\Omega)$.

Theorems & Definitions (24)

  • Remark 2.1: Hanging nodes
  • Lemma 3.1: Lifting error
  • proof
  • Corollary 3.2: Control of the non-confomity
  • Lemma 4.1: Aubin-Nitsche
  • proof
  • Theorem 4.2: A priori estimate
  • Remark 4.3: Higher-order term
  • proof
  • Lemma 5.1: Aubin-Nitsche
  • ...and 14 more