Implicit numerical approach for biological Schnakenberg model employing finite element method (FEM)

Abstract

Finite Element Method (FEM) based numerical techniques for two-dimensional and two-directional problems are challenging for numerical analysis practitioners. In this study, an implicit numerical scheme was developed to solve a set of coupled non-linear partial differential equations using the finite element method. This study presents a numerical approach using Finite Element Method (FEM) to implicitly solve the Schnakenberg biological model. This method uses the inherent advantages of the finite element method, such as its adaptability and ability to deal with complex geometries, while also introducing a strategy to improve stability. A system of algebraic equations is derived from the model equations to compute spatiotemporal dynamics by discretizing the spatial domain. The numerical solution is obtained by iteratively solving the resulting algebraic equations, by employing suitable linear solvers and convergence criteria. The results also demonstrate that this method captures intricate spatial patterns of morphogenesis concentration in animal species. Furthermore, the objective of this study is to analyze stripe and spot-like patterns during morphogenesis in animal species.