Error Analysis of the Galerkin Finite Element Method for the Gray-Scott Model

This paper proposes an extensive finite element analysis of the one-dimensional Gray-Scott reaction-diffusion model (GSRDM), which is a fundamental framework for examining pattern formation in chemical and biological phenomena. A fully discrete numerical strategy was constructed utilizing the Galerkin finite element method (GFEM) for spatial discretization and the backward Euler (BE) scheme for temporal discretization. The nonlinear term is meticulously treated using a fully discrete formulation, preserving its authentic characteristics. The stability and convergence analysis of the discrete formulation is rigorously investigated by employing the error splitting and elliptic projection techniques with a special treatment for the nonlinear reaction terms. Numerical investigations employing a MATLAB script validated the predicted convergence rates and affirmed the precision of the proposed techniques. The impact of space and time-step refinements is examined comprehensively, supported by exact solutions and norm-based error analysis. A comparison with referenced works is discussed to demonstrate the effectiveness of the proposed scheme. This research offers a robust and adaptable framework for further studies on nonlinear reaction-diffusion systems.