A Numerical Scheme Based on Finite Element Method for Solving Two-Dimensional Time-Dependent Convection-Diffusion- Reaction Equation

In this study, a numerical scheme is developed for solving a two-dimensional, time-dependent linear convection-diffusion-reaction (CDR) equation with spatially and temporally varying coefficients. The scheme employs the Galerkin finite element method for spatial discretization and the implicit Euler method for time integration. To solve the algebraic system arising from the discretization, the matrix inverse method was applied in conjunction with sparse matrix techniques. Finite element discretization is performed using four-node rectangular elements. A brief analysis is presented to demonstrate the stability and convergence of the proposed scheme. Test examples are provided to illustrate the performance of the method. For the cases considered, the numerical solutions obtained by the scheme show good agreement with the exact solutions. Moreover, the finite element method provides accuracy comparable to the central finite difference method and offers greater flexibility in handling problems with varied or complex boundary conditions. The computational strategy presented here gives valuable insight into the application of the finite element method to partial differential equations arising in various physical phenomena.