Comparative Study of Finite Difference and Finite Element Methods for Solving 2D Parabolic Convection–Diffusion–Reaction Equations With Variable Coefficients

This study investigates the numerical solution of two-dimensional parabolic convection–diffusion–reaction (CDR) equations with variable coefficients using the finite difference method (FDM) and the finite element method (FEM). The FDM employs central differences for spatial discretization and the implicit Euler method for time integration, while the FEM uses the Galerkin approach with rectangular elements and three-point Gauss–Legendre quadrature for spatial integrals, followed by implicit Euler discretization in time. Three test problems are considered to compare accuracy and efficiency. For small diffusion coefficients, the FEM provides higher accuracy, whereas for larger diffusion coefficients, both methods deliver nearly identical accuracy. Despite its improved accuracy in certain cases, the FEM typically involves a higher computational cost than the FDM. Based on the results, the study recommends the use of FEM for problems with boundary or interior layers.