Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction-diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2-D reaction-diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction-diffusion systems to find unknown concentrations(u,v)in element interiors and boundaries in the weak Galerkin finite element spaceWG(P-0, P-0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction-diffusion systems, stability analysis and error bounds for semi-discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well-known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.