An Error Analysis of the CN Weighed DG θ Method of the Convection Equation

Muhammad Shakhawat Hossain ,Chunguang Xiong

doi:10.3390/math9090970

Abstract

In this research paper, a weighed DG finite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general finite element meshes. The efficiency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and flexibility of the method.

Highlights

Convection plays an important role in the application of meteorology, gas dynamics, oceanography, weather forecasting, aeroacoustics, turbulent flows, turbomachinery, oil recovery simulation, modeling of shallow water, the transport of contaminants in porous media, viscoelastic flows, magneto-hydrodynamics, and electro-magnetism
The discontinuous Galerkin finite element method (DGFEM) was introduced by Reed and Hill [1] in 1973 for solving the neutron transport equation. They compared their method with the continuous Galerkin finite element method by using numerical experiments and showed worthwhile stability properties of DGFEM
Xiong C., Luo F., and Ma X. et al in [19] derived the a priori error analysis for the streamline diffusion discontinuous Galerkin finite element approximation of the optimal distributed control problem governed by the first-order linear hyperbolic equation

Summary

Introduction

Convection plays an important role in the application of meteorology, gas dynamics, oceanography, weather forecasting, aeroacoustics, turbulent flows, turbomachinery, oil recovery simulation, modeling of shallow water, the transport of contaminants in porous media, viscoelastic flows, magneto-hydrodynamics, and electro-magnetism. Xiong and Li in [18] clarified the convergence properties of the optimal control problem governed by convection-diffusion equations They extended the a posteriori error estimates and the a priori error estimates for both states, adjoints, and the control variable approximation. Xiong C., Luo F., and Ma X. et al in [19] derived the a priori error analysis for the streamline diffusion discontinuous Galerkin finite element approximation of the optimal distributed control problem governed by the first-order linear hyperbolic equation. In this article, they analyzed the behavior of the error for h tending to zero and the degree p tending to infinity. We believe that the present way provides a simpler and more elegant error approximation of the convection equation for both space and time discretization

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Conclusion