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A Péclet-robust discontinuous Galerkin method for nonlinear diffusion with advection

Lourenço Beirão da Veiga, Daniele A. Di Pietro, Kirubell B. Haile

TL;DR

The paper tackles the challenge of accurately approximating a nonlinear diffusion–advection–reaction problem with a Discontinuous Galerkin method. It introduces a Péclet-robust scheme that discretizes the diffusion term via a full gradient with jump liftings and interior penalties, while stabilizing the advection term with a strengthened upwind approach, all within a local, elementwise Pe framework. The main contribution is a set of Pe-dependent, fully local error estimates for Sobolev indices $p∈(1,∞)$, capturing diffusion- and advection-dominated regimes and recovering classical linear ($p=2$) results; the analysis also extends to nonconforming settings via new face Pe concepts. Numerical tests validate the theory, showing correct pre-asymptotic behavior and highlighting the method’s robustness in advection-dominated regions, with implications for stable, pressure-robust, and advection-robust schemes in non-Newtonian flow contexts.

Abstract

We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1, \infty)$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local Péclet numbers. A set of numerical tests supports the theoretical derivations.

A Péclet-robust discontinuous Galerkin method for nonlinear diffusion with advection

TL;DR

The paper tackles the challenge of accurately approximating a nonlinear diffusion–advection–reaction problem with a Discontinuous Galerkin method. It introduces a Péclet-robust scheme that discretizes the diffusion term via a full gradient with jump liftings and interior penalties, while stabilizing the advection term with a strengthened upwind approach, all within a local, elementwise Pe framework. The main contribution is a set of Pe-dependent, fully local error estimates for Sobolev indices , capturing diffusion- and advection-dominated regimes and recovering classical linear () results; the analysis also extends to nonconforming settings via new face Pe concepts. Numerical tests validate the theory, showing correct pre-asymptotic behavior and highlighting the method’s robustness in advection-dominated regions, with implications for stable, pressure-robust, and advection-robust schemes in non-Newtonian flow contexts.

Abstract

We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and -type diffusion, with Sobolev indices . The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local Péclet numbers. A set of numerical tests supports the theoretical derivations.
Paper Structure (19 sections, 7 theorems, 80 equations, 4 figures)

This paper contains 19 sections, 7 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. The discrete setting and preliminary results
  4. Local and broken spaces
  5. Trace operators and integration by parts formula
  6. Jump liftings and discrete gradient
  7. Discrete problem and main results
  8. Discrete problem
  9. Main results
  10. Dimensionless numbers and reference quantities
  11. Norms
  12. Error estimate
  13. Theoretical analysis
  14. Properties of the diffusion function
  15. Properties of the advection-reaction bilinear form
  16. ...and 4 more sections

Key Result

Lemma 1

Let $p \in (1,\infty)$ and an integer $n \ge 1$ be given. For all $x,y,z \in \mathbb{R}^n$ and any real number $\delta > 0$, it holds with $c(\delta)$ positive constant depending only on $p$ and $\delta$.

Figures (4)

  • Figure 1: Example of Section \ref{['sec:ex1']}. Convergence rate of $\mathrm{ERR}_h$ in the diffusion-dominated regime. Theoretical convergence rate: $\frac{kp}{2}$ for $p\leq 2$ and $\frac{kp'}{2}$ for $p>2$.
  • Figure 2: Example of Section \ref{['sec:ex1']}. Convergence rate of $\mathrm{ERR}_h$ in the advection-dominated regime. Theoretical convergence rate: $k+\frac{1}{2}$ for all $p$.
  • Figure 3: Example of Section \ref{['sec:ex2']} with exponential solution. Convergence rate of $\mathrm{ERR}_h$. Theoretical convergence rate: $\frac{kp}{2}$ for $p \in \{1.5, 1.75 \}$.
  • Figure 4: Example of Section \ref{['sec:ex2']} with polynomial solution. Convergence rate of $\mathrm{ERR}_h$. Theoretical convergence rate: $\frac{kp}{2}$ for $p \in \{1.5, 1.75 \}$.

Theorems & Definitions (19)

  • Lemma 1: Modified monotonicity of the power flux function
  • proof
  • Remark 2: Polytopal meshes
  • Lemma 3: Properties of the jump lifting
  • proof
  • Lemma 4: Approximation properties of the discrete gradient
  • proof
  • Remark 5: Local boundedness of $G_h^{k}\circ\pi_{h}^{k}$
  • Remark 6: Generalizations
  • Theorem 7: Convergence
  • ...and 9 more