An elemental scale adjustment method for the finite element and finite difference solutions of diffusion-reaction and convection–diffusion equations

A method for the stable-accurate solution of convection–diffusion and diffusion-reaction equations is proposed to produce a solution similar to highly refined meshes for any level of discretisation. The method is applied on finite element and finite difference methods. The idea is based on the analytical or numerical solutions of the governing equation in a test domain and then determining the level of adjustment in an element with length l by solving the global system of equations in the test domain. The adjustment can be done either on the coefficients of the equation or enrichment of normal shape functions in the Galerkin finite element scheme. The numerical experiments are performed in one and two dimensional cases. Different mesh schemes and boundary conditions are used. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Damköhler and Peclet numbers.

► A new enrichment for the numerical solution of transport equations.
► The method is flexible such that can be applied on different numerical schemes.
► It provides a multiscale solution scheme in the Galerkin finite element method.