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Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms

Alexandre Ern, Iain Smears, Martin Vohralík

Abstract

We establish the existence of liftings into discrete subspaces of $\mathbf{H}(\mathrm{div})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.

Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms

Abstract

We establish the existence of liftings into discrete subspaces of of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.

Paper Structure

This paper contains 16 sections, 7 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Discrete $p$-robust $\bm{H}(\mathop{\mathrm{div}}\nolimits)$-liftings
  3. A posteriori error analysis of problems with $H^{-1}$ source terms
  4. Setting
  5. Flux equilibration for elliptic problems with $H^{-1}$ source terms
  6. Proof of Theorem \ref{['thm:aposteriori']}
  7. Proof of the guaranteed upper bound \ref{['eq:upper_bound']}
  8. Proof of the local efficiency \ref{['eq:local_efficiency']}
  9. Proof of the global efficiency \ref{['eq:global_efficiency']}
  10. Proof of Theorem \ref{['thm:flux_stability']}
  11. Proof of Theorem \ref{['thm:flux_stability_2']}
  12. Construction of $\bm{\widetilde{\sigma}}_h$
  13. Construction of $\bm{\sigma}_h^{\mathrm{c}}$
  14. Admissibility of $\bm{\sigma}_h^{\dagger}$
  15. Proof of \ref{['eq:flux_stability_2']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p\geq 1$. For any $f \in \mathcal{P}_{p-1}(\mathcal{T})$ and $\bm{\xi} \in \bm{RTN}_{p-1}(\mathcal{T})$, satisfying $(f,1)=0$ if $\Gamma_{\mathrm N}=\partial\Omega$, we have

Figures (1)

  • Figure 1: Original equilibration patches of reference DolejsiErnVohralik2016 (left) and extended equilibration patches necessary for estimates robust with respect to an arbitrary number of levels of hanging nodes (right)

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • ...and 1 more