A modified diffusion coefficient technique for the convection diffusion equation

A new modified diffusion coefficient (MDC) technique for solving convection diffusion equation is proposed. The Galerkin finite-element discretization process is applied on the modified equation rather than the original one. For a class of one-dimensional convection–diffusion equations, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. The application of the derived analytic formula of MDC is extended for other classes of convection–diffusion equations, where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction. The numerical results of the proposed approach for two-dimensional, variable coefficients, with boundary layers, convection-dominated problems show stable and accurate solutions even on coarse grids. Accordingly, multigrid based solvers retain their efficient convergence rates.