Numerical solution of convection–diffusion–reaction equations by a finite element method with error correlation

Sadia Akter Lima,Md Kamrujjaman,Md Shafiqul Islam

doi:10.1063/5.0050792

Sadia Akter Lima, Md Kamrujjaman + Show 1 more

Open Access

https://doi.org/10.1063/5.0050792

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Journal: AIP Advances	Publication Date: Aug 1, 2021
Citations: 14	License type: cc-by

Affiliation: University of Dhaka, University of Calgary

Abstract

This study contemplates the Finite Element Method (FEM), a well-known numerical method, to find numerical approximations of the Convection–Diffusion–Reaction (CDR) equation. We concentrate on analyzing the convergence and stability of the nonlinear parabolic partial equations. The method is generally applied without truncating the nonlinear terms and avoiding restrictive assumptions. Regular and irregular geometrical shapes are the key objective of this research paper. This study also focuses on the accuracy and acceptance of the FEM method by utilizing dissipation error, dispersion error, and total error analysis. The results are portrayed both graphically and in a tabular form, which virtually ensures the method’s validity and the algorithm’s efficiency to sustain the accuracy, simplicity, and applicability for solving nonlinear CDR equations. The proposed technique may also be applied for solving any nonlinear reaction–diffusion equations.

Highlights

Modeling real life and industrial problems by applying partial differential equations (PDEs) is challenging for researchers and scientists
The Finite Difference Method (FDM) has been used widely despite some limitations; by using this method, we can get the solutions at particular grid points, but it cannot obtain the solutions at every single point between two grid points
It is very challenging to find an analytical solution to problems that develop from real-life issues; numerical methods can significantly help us obtain approximate solutions

Summary

INTRODUCTION

Modeling real life and industrial problems by applying partial differential equations (PDEs) is challenging for researchers and scientists. Researchers have attempted to solve these problems analytically or numerically using different methods and equations to obtain higher accuracy levels. The drawback is the computational cost to achieve higher accuracy To overcome these obstacles, for some time, the Galerkin Finite Element Method (GFEM) played the most important role in solving engineering and industrial problems, including complicated geometries and material properties. The second drawback is the computational cost to obtain higher accuracy To overcome these obstacles, the FEM becomes the most popular numerical method for solving physical and biological problems. Islam et al. explored numerical solutions of the initial value problem (IVP) using the finite element method with the Taylor series They introduced an integration technique to approximate the numerical solution of an IVP of differential equations. II, we discuss the formulation of the FEM for the nonlinear CDR equation

FORMULATION OF THE FEM FOR THE CDR EQUATION

CONVERGENCE AND STABILITY ANALYSIS

QUANTIFICATION OF ERRORS

OF EXAMPLES AND RESULTS

CONCLUSION