Métodos de elementos finitos híbridos aplicados na solução de equações transientes de reação-difusão

Abstract

Systems of reaction-diffusion partial differential equations are characterized by the ability to produce a wide variety of complex spatial patterns. The models are often applied in different areas of knowledge such as chemistry, physics, biology, among others. Due to the nonlinearity of the problem present in most applications, reaction-diffusion models have often been studied in numerical and computational terms. In this context, the present work aims at the application of hybrid finite element methods for the spatial approximation combined with SBDF (Semi-Implicit Backward Differentiation Formulas) methods for the time discretization to build numerical approximations for reaction-diffusion problems. Hybrid methods are characterized by imposing continuity through the use of a Lagrange multiplier. In this approach the treatment of high-order numerical approximations becomes less complex and computationally efficient. SBDF methods, on the other hand, are highorder alternatives for time integration and allow treating the non-linear reaction term explicitly. Numerical experiments with the proposed formulation were carried out to show the optimal convergence rates in space and time. In addition, simulations of cardiac electrophysiology models and of the Gray-Scott model were performed satisfactorily with the proposed methodology.