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Quasi-optimal Discontinuous Galerkin discretisations of the $p$-Dirichlet problem

J. Blechta, P. A. Gazca-Orozco, A. Kaltenbach, M. Růžička

Abstract

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of -minimal regularity-, since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with $(p,δ)$-structure that only assume the natural $W^{1,p}$-regularity of the exact solution, and which do not contain any oscillation terms.

Quasi-optimal Discontinuous Galerkin discretisations of the $p$-Dirichlet problem

Abstract

The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of -minimal regularity-, since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with -structure that only assume the natural -regularity of the exact solution, and which do not contain any oscillation terms.
Paper Structure (16 sections, 18 theorems, 128 equations, 1 figure, 12 tables)

This paper contains 16 sections, 18 theorems, 128 equations, 1 figure, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of the non-linear operator
  4. Mesh regularity
  5. Broken function spaces and projectors
  6. DG gradient and jump operator
  7. Construction of moment-preserving operators
  8. Primal formulation
  9. Continuous primal formulation
  10. Discrete primal formulation
  11. Mixed formulation
  12. Continuous mixed formulation
  13. Discrete mixed formulation
  14. Numerical experiments
  15. Primal formulation
  16. ...and 1 more sections

Key Result

Proposition 2.3

\newlabellem:hammer0 Let $\pmb{\mathsf{\mathcal{S}}}$ have $(p,\delta)$-structure, let $\varphi$ be defined in eq:def_phi, and let $\pmb{\mathsf{\mathcal{F}}},\pmb{\mathsf{\mathcal{F}}}^*$ be defined in eq:def_F. Then, uniformly with respect to $\pmb{\mathsf{Q}}, \pmb{\mathsf{P}} \in \mathbb{R}^{n The constants in eq:hammeraeq:F-F*3 depend only on the characteristics of ${\pmb{\mathsf{\mathcal{S}

Figures (1)

  • Figure 1: Initial mesh $\mathcal{T}_{h_0}$

Theorems & Definitions (43)

  • Definition 2.1: $(p,\delta)$-structure
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Lemma 2.5: Change of shift
  • Proposition 3.1: Interpolation estimate
  • Proof 1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 33 more