A modified diffusion coefficient technique for the convection diffusion equation

A new modified diffusi on coefficient (MDC) technique for solv ing conve ction 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 convec-tion–diffusion equations, we derive the modi fied 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. The convection–diffusion equation s play an important role in many engineering and physics phenomena. In most practical problems , the magnitude of convection coefficient is much greater than that of diffusion coefficient. So, these problems are called convection-dominated or singularl y-perturbed. The numerical solution of these problems represents serious difficulties because the solution of these diffusion–convection problems possess boundary layers that are small sub-regions, where derivatives of the solution are very large. These boundary layers make standard finite-element or finite-difference methods unsuitable for solving these problems. This is because the numerica l solutions produce non-physical oscillatio ns and low-order of accuracy unless refined meshes are introduced in the boundary regions using an adaptive mesh-refinement strategy. For this strategy to be effective, it is important that the error does not propagat e into regions where refinement is not needed. So, the computational costs increase to obtain satisfied numerical results. Another difficulty occurs when multigrid is used for solving convection-dominated problems using classical discretiza-tion methods. Even if the grid where the solution is computed provides suitable accuracy, the multigrid algorithm requires a sequence of coarser grids, and it is important that the discretizatio ns on these grids capture the character of the solution with a reasonable degree of accuracy. For these reasons, it is necessary to have a discretization strategy that does not have the deficiencies exhibited by the classical discretization method.