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

Abstract

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 space WG(P 0 , P 0 , RT 0). 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.

Simulation of predator–prey system with two-species, two chemicals and an additional chemotactic influence via direct meshless local Petrov–Galerkin method

“…A system of diffusion-reaction equations with application in biology is: where f and F are linear and nonlinear functions, respectively. There are several research works that they numerically investigated model (1.1), for example, finite difference method (Dillon et al , 1994), finite volume method (Shakeri and Dehghan, 2011), finite element method (Barreira et al , 2011; Khaled-Abad and Salehi, 2021; Madzvamuse et al , 2005), element free Galerkin method (Dehghan and Abbaszadeh, 2016; Dehghan et al , 2016), Chebyshev spectral method (Tehseen Saleem and Ali, 2018), Legendre spectral element method (Dehghan and Sabouri, 2013), spectral meshless radial point interpolation approach (Shivanian and Jafarabadi, 2020), local radial basis function (RBF) (Sarra, 2012) and convolutional neural network (Zhu and He, 2022). There are various versions of the predator–prey system, for instance, the following Gause-type model (Gause et al , 1936): where ∏( t ) and u ( t ) denote the population size of prey and predator spaces, respectively.…”