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High order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

Rodolfo Bermejo, Manuel Colera

TL;DR

This work introduces high-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation on time-evolving meshes, employing Backward Differentiation Formulas up to order $q=5$. By formulating a nearly-conservative weak form on the transported domain and using high-order quadrature for moving-domain integrals, the approach achieves stability and optimal convergence across diffusive and advection-dominated regimes without exponential dependence on $\mu^{-1}$. The analysis combines moving-mesh finite elements with curved-element theory to establish well-posedness, error bounds, and regime-sensitive rates, while numerical experiments confirm near mass conservation and the expected $L^2$-convergence behavior, with higher-order schemes offering improved accuracy and conservation properties. These results are relevant for accurate, stable simulations of advection-diffusion processes in regimes ranging from diffusion-dominated to highly transport-dominated flows. The methodology thus provides a robust, high-order alternative to traditional Eulerian and purely Lagrangian schemes in convection-d-diffusion problems with mass conservation considerations.

Abstract

We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $μ$ is large, the error is $O(h^{k+1}+Δt^{q})$, whereas in the advective regime, $μ\ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{Δt})+Δt^{q})$. It is worth remarking that the error constant does not have exponential $μ^{-1}$ dependence.

High order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

TL;DR

This work introduces high-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation on time-evolving meshes, employing Backward Differentiation Formulas up to order . By formulating a nearly-conservative weak form on the transported domain and using high-order quadrature for moving-domain integrals, the approach achieves stability and optimal convergence across diffusive and advection-dominated regimes without exponential dependence on . The analysis combines moving-mesh finite elements with curved-element theory to establish well-posedness, error bounds, and regime-sensitive rates, while numerical experiments confirm near mass conservation and the expected -convergence behavior, with higher-order schemes offering improved accuracy and conservation properties. These results are relevant for accurate, stable simulations of advection-diffusion processes in regimes ranging from diffusion-dominated to highly transport-dominated flows. The methodology thus provides a robust, high-order alternative to traditional Eulerian and purely Lagrangian schemes in convection-d-diffusion problems with mass conservation considerations.

Abstract

We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order . The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter is large, the error is , whereas in the advective regime, , the convergence is . It is worth remarking that the error constant does not have exponential dependence.
Paper Structure (8 sections, 18 theorems, 225 equations, 8 figures)

This paper contains 8 sections, 18 theorems, 225 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The nearly-conservative Lagrange-Galekin-(BDF-q) methods
  3. On the partitions $\mathcal{T}_{h}$ and$\ \mathcal{T}_{h}(t)$ and their associated finite element spaces $V_{h}$ and $V_{h}(t)$ respectively
  4. Calculation of the integrals $\left( c_{h}^{n-i},v_{h}^{n-i}\right) _{\Omega _{h}(t_{n-i})}$ of (\ref{['s2:11']}) for $1\leq i\leq q$
  5. Stability and error analysis
  6. Stability analysis
  7. Error Analysis
  8. Numerical experiments

Key Result

Lemma 1

Assume that $u\in C$$([0,T];W^{k+1,\infty }(\Omega )^{d}),\ k\geq 0$. Then for all $n$ such that $q\leq$$n\leq N$, there exists a unique solution $t\rightarrow X(x,t_{n};t)\ (t\in \lbrack t_{n-q},t_{n})\subset \lbrack 0,T])$ of (int2) such that $X(x,t_{n};t)\in C^{1}(0,T;W^{k+1,\infty }(\Omega )^{d}

Figures (8)

  • Figure 1: The straight simplex $\overline{T}_{l}$ is the affine image of the simplex $\widehat{T}$, whereas the blue curved simplex $T_{l}=F_{l}(\widehat{T})$.
  • Figure 2: The green curved element $T_{l}(t) \in \mathcal{T}_{h}(t)$ may intersect several elements (blue) of the fixed partition $\mathcal{T}_{h}$.
  • Figure 3: The blue simplex $T_{l}$ belongs to fixed mesh $\mathcal{T}_{h}$; the red simplex $T_{l}(t)$ belong to the mesh $\mathcal{T}_{h}(t)$ and is the image of the blue simplex $T_{l}$ by the mapping $X(x,t_{n};t)$; the green simplex $\widetilde{T}_{l}(t)$ is the isoparametric simplex that approximates the red element $T_{l}(t)$.
  • Figure 4: The band of width $2r_{\partial \Omega}$ about $\Omega_{h}(t)$.
  • Figure 5: Numerical solution.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Lemma 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • Remark 10
  • ...and 21 more