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A modified dynamic diffusion finite element method with optimal convergence rate for convection-diffusion-reaction equations

Shaohong Du, Qianqian Hou, Xiaoping Xie

TL;DR

The paper develops a stabilization-parameter-free nonlinear dynamic diffusion finite element method for the convection-diffusion-reaction equation, ensuring oscillation-free solutions in convection-dominated regimes. It introduces a bounded artificial diffusion operator tied to the residual and a two-scale finite element space, proving existence and, under a small-mesh restriction, uniqueness of the discrete solution. An optimal first-order convergence rate in the energy norm (plus a dissipative term) is established, with numerical experiments corroborating the theory and showing robust performance even with boundary layers. The approach unifies diffusion mechanisms for general velocity fields and variable reaction coefficients, offering a stabilization-free alternative with practical convergence guarantees.

Abstract

In this paper, we develop a modified nonlinear dynamic diffusion (DD) finite element method for convection-diffusion-reaction equations. This method is free of stabilization parameters and is capable of precluding spurious oscillations. We prove existence and, under an assumption of small mesh size, uniqueness of the discrete solution, and derive the optimal first order convergence rate of the approximation error in the energy norm plus a dissipation term. Numerical examples are provided to verify the theoretical analysis.

A modified dynamic diffusion finite element method with optimal convergence rate for convection-diffusion-reaction equations

TL;DR

The paper develops a stabilization-parameter-free nonlinear dynamic diffusion finite element method for the convection-diffusion-reaction equation, ensuring oscillation-free solutions in convection-dominated regimes. It introduces a bounded artificial diffusion operator tied to the residual and a two-scale finite element space, proving existence and, under a small-mesh restriction, uniqueness of the discrete solution. An optimal first-order convergence rate in the energy norm (plus a dissipative term) is established, with numerical experiments corroborating the theory and showing robust performance even with boundary layers. The approach unifies diffusion mechanisms for general velocity fields and variable reaction coefficients, offering a stabilization-free alternative with practical convergence guarantees.

Abstract

In this paper, we develop a modified nonlinear dynamic diffusion (DD) finite element method for convection-diffusion-reaction equations. This method is free of stabilization parameters and is capable of precluding spurious oscillations. We prove existence and, under an assumption of small mesh size, uniqueness of the discrete solution, and derive the optimal first order convergence rate of the approximation error in the energy norm plus a dissipation term. Numerical examples are provided to verify the theoretical analysis.

Paper Structure

This paper contains 15 sections, 6 theorems, 104 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Notation and weak formulation
  3. A modified DD method
  4. Existence and uniqueness of approximation solution
  5. Preliminary results
  6. Proof:
  7. Proof:
  8. Existence and uniqueness of approximation solution
  9. Proof:
  10. Proof:
  11. Optimal estimate
  12. Proof:
  13. Numerical experiments
  14. Fixed point iteration
  15. Numerical examples

Key Result

Lemma 4.1

For any $u_{hb},v_{hb}, w_{hb}\in V_{hb}^{D}$, there hold and where and $M_{1}$ is the same as in formula5.

Figures (2)

  • Figure 1: The domain $\Omega$ and $N\times N$ meshes with $N=2,\ 4,\ 8$.
  • Figure 2: Approximation solution $u_{hb}$ over meshes $8\times8$ (left) and $16\times16$ (right) with $\varepsilon=10^{-6}, \sigma=1$.

Theorems & Definitions (14)

  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Remark 5.1
  • ...and 4 more