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A weak Galerkin finite element method for nonlinear convection-diffusion equation

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  • Li, Wenjuan
  • Gao, Fuzheng
  • Cui, Jintao

Abstract

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.

Suggested Citation

  • Li, Wenjuan & Gao, Fuzheng & Cui, Jintao, 2024. "A weak Galerkin finite element method for nonlinear convection-diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 461(C).
    • Handle: RePEc:eee:apmaco:v:461:y:2024:i:c:s0096300323004848
    DOI: 10.1016/j.amc.2023.128315
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    References listed on IDEAS

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    1. Zhang, Tie & Chen, Yanli, 2019. "An analysis of the weak Galerkin finite element method for convection–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 612-621.
    2. Guo, Yan & Shi, Yu-feng & Li, Yi-min, 2016. "A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 172-185.
    3. Wang, Haijin & Shu, Chi-Wang & Zhang, Qiang, 2016. "Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 237-258.
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