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Error analysis of a Euler finite element scheme for Natural convection model with variable density

Li Hang, Chenyang Li

TL;DR

This work addresses the error analysis of a first-order Euler time discretization coupled with finite element spatial discretization for the natural convection model with variable density (NCVD). By introducing the equivalent formulation with $\sigma=\sqrt{\rho}$, the authors obtain unconditional stability via discrete energy inequalities and prove $L^2$-norm error estimates for density, velocity, and temperature under regularity assumptions and a time-step constraint $\tau\le C h^2$, achieving an overall convergence rate of $O(\tau + h^2)$. The fully discrete scheme employs a mini element for velocity-pressure, $P_1$-elements for density and temperature, and Raviart–Thomas spaces for $H(\mathrm{div})$-conforming velocity, with a postprocessed velocity to ensure stability. Numerical tests using a manufactured solution in 2D/3D validate the theoretical results, showing expected first-order temporal accuracy and second-order spatial accuracy under appropriate time stepping, while demonstrating the method’s efficiency for NCVD problems.

Abstract

In this paper, we derive first-order Euler finite element discretization schemes for a time-dependent natural convection model with variable density (NCVD). The model is governed by the variable density Navier-Stokes equations coupled with a parabolic partial differential equation that describes the evolution of temperature. Stability and error estimate for the velocity, pressure, density and temperature in $L^2$-norm are proved by using finite element approximations in space and finite differences in time. Finally, the numerical results are showed to support the theoretical analysis.

Error analysis of a Euler finite element scheme for Natural convection model with variable density

TL;DR

This work addresses the error analysis of a first-order Euler time discretization coupled with finite element spatial discretization for the natural convection model with variable density (NCVD). By introducing the equivalent formulation with , the authors obtain unconditional stability via discrete energy inequalities and prove -norm error estimates for density, velocity, and temperature under regularity assumptions and a time-step constraint , achieving an overall convergence rate of . The fully discrete scheme employs a mini element for velocity-pressure, -elements for density and temperature, and Raviart–Thomas spaces for -conforming velocity, with a postprocessed velocity to ensure stability. Numerical tests using a manufactured solution in 2D/3D validate the theoretical results, showing expected first-order temporal accuracy and second-order spatial accuracy under appropriate time stepping, while demonstrating the method’s efficiency for NCVD problems.

Abstract

In this paper, we derive first-order Euler finite element discretization schemes for a time-dependent natural convection model with variable density (NCVD). The model is governed by the variable density Navier-Stokes equations coupled with a parabolic partial differential equation that describes the evolution of temperature. Stability and error estimate for the velocity, pressure, density and temperature in -norm are proved by using finite element approximations in space and finite differences in time. Finally, the numerical results are showed to support the theoretical analysis.

Paper Structure

This paper contains 13 sections, 6 theorems, 156 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An equivalent system
  4. Finite element approximations
  5. Finite element discretization scheme
  6. Stability Result
  7. Error Estimate
  8. Estimate of $\| \textbf{e}^{n+1}_{\textbf{u},h}\|^2_{L^2}$
  9. Estimate of $\| e^{n+1}_{\theta,h}\|^2_{L^2}$
  10. Main result
  11. Numerical results
  12. Conclusions
  13. Acknowledgments

Key Result

Lemma 2.1

Let $a_k, b_k$ and $\gamma_k$ be the nonnegative numbers such that Suppose $\tau\gamma_k<1$ and set $\sigma_k=(1-\tau\gamma_k)^{-1}$. Then there holds

Figures (4)

  • Figure 1: Numerical solutions of velocity at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0.
  • Figure 2: Numerical solutions of pressure at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0.
  • Figure 3: Numerical solutions of density at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0.
  • Figure 4: Numerical solutions of temperature at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0.

Theorems & Definitions (13)

  • Lemma 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 3 more