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Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes

Bernardo Cockburn, Alexandre Ern, Rekha Khot

TL;DR

The paper addresses improved $L^2$-error estimates for the wave equation discretized with hybrid nonconforming methods on simplicial meshes. It introduces the $\text{H}$-interpolation operator to achieve superclose or optimal approximation properties, and develops energy and duality arguments to obtain a time-integrated $L^2$-error bound of order $\mathcal{O}(h^{k+1+s})$, where $s\in(\tfrac{1}{2},1]$. The analysis encompasses equal- and mixed-order settings, allows for relaxed regularity on the dual divergence, and extends the stabilization framework beyond plain Least-Squares. The results enhance theoretical guarantees for HDG/HHO/WG discretizations of the wave equation and may inform post-processing or post hoc improvements in practice, particularly regarding the time-integrated primal variable.

Abstract

We present improved $L^2$-error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous Galerkin, the hybrid high-order or the weak Galerkin methods. We consider both equal-order and mixed-order settings on simplices, and include the lowest-order case with piecewise constant unknowns on the faces and in the cells. Our main result is a superclose, resp., optimal bound on the above error in the equal-, resp., mixed-order case. A key result of independent interest to achieve these estimates are novel approximation estimates for an interpolation operator inspired from the hybridizable discontinuous Galerkin literature.

Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes

TL;DR

The paper addresses improved -error estimates for the wave equation discretized with hybrid nonconforming methods on simplicial meshes. It introduces the -interpolation operator to achieve superclose or optimal approximation properties, and develops energy and duality arguments to obtain a time-integrated -error bound of order , where . The analysis encompasses equal- and mixed-order settings, allows for relaxed regularity on the dual divergence, and extends the stabilization framework beyond plain Least-Squares. The results enhance theoretical guarantees for HDG/HHO/WG discretizations of the wave equation and may inform post-processing or post hoc improvements in practice, particularly regarding the time-integrated primal variable.

Abstract

We present improved -error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous Galerkin, the hybrid high-order or the weak Galerkin methods. We consider both equal-order and mixed-order settings on simplices, and include the lowest-order case with piecewise constant unknowns on the faces and in the cells. Our main result is a superclose, resp., optimal bound on the above error in the equal-, resp., mixed-order case. A key result of independent interest to achieve these estimates are novel approximation estimates for an interpolation operator inspired from the hybridizable discontinuous Galerkin literature.

Paper Structure

This paper contains 31 sections, 16 theorems, 144 equations, 1 table.

Table of Contents

  1. Introduction
  2. Model problem and space semi-discretization
  3. Model problem
  4. Space semi-discretization
  5. Mesh and polynomial spaces
  6. HHO/WG space semi-discretization
  7. HDG rewriting
  8. The H-interpolation operator
  9. Main results
  10. Main results on the H-interpolation operator
  11. Well-posedness and approximation estimates
  12. Extension to weaker regularity of dual variable
  13. A priori error analysis for the wave equation
  14. Errors and their equations
  15. Error estimates
  16. ...and 16 more sections

Key Result

Lemma 3.1

For all $\ul{v}:=(\boldsymbol{\sigma},v)\in \boldsymbol{H}^{\nu_{\boldsymbol{\sigma}}}(T)\times H^{\nu_v}(T)$ with $\nu_{\boldsymbol{\sigma}}>\frac{1}{2}$ and $\nu_v>\frac{1}{2}$, the $\rm{H}$-interpolation operator $\ul{\Pi}^{\textsc{h}}_T$ is well-defined.

Theorems & Definitions (36)

  • Remark 2.1: Role of stabilization
  • Remark 2.2: Initial face values
  • Remark 2.3: Comparison with du2019invitation
  • Lemma 3.1: Well-posedness
  • Theorem 3.2: Approximation estimates
  • Remark 3.3: Supercloseness/optimality
  • Remark 3.4: Approximation estimates
  • Remark 3.5: Separation of primal and dual variables
  • Remark 3.6: Relaxed regularity
  • Theorem 3.7: Approximation estimates with negative regularity on $\nabla{\cdot}\boldsymbol{\sigma}$
  • ...and 26 more