Métodos de elementos finitos híbridos estabilizados para a equação de Cahn-Hilliard e suas aplicações

Abstract

Several interface problems demand the numerical solution of partial differential equations in moving domains, where the movements of the interfaces are unknown and difficult to calculate when they undergo topological changes. The phase field approach has been shown to be a powerful tool for modeling such problems, considering a known and fixed computational domain. In this context, the Cahn-Hilliard equation, initially developed to model the separation of binary alloys, has been widely used in several applications ranging from modeling tumor growth to image processing. It consists of a non-linear fourth-order parabolic partial differential equation that presents great challenges in the numerical solution, which in certain cases can present non-physical oscillations and demand the use of extremely refined meshes and time steps. This work aims to overcome such numerical difficulties through hybrid finite element formulations in space and second-order formulations in time aiming at robustness and efficiency. The classical Cahn-Hilliard equation as well as other models based on it are studied from a numerical point of view to verify the order of convergence of the presented methods and evaluate their efficiency and precision. In particular, some applications of the Cahn-Hilliard equation such as in the modeling of tumor growth and the electrowetting process are also considered in this work.

Several interface problems demand the numerical solution of partial differential equations in moving domains, where the movements of the interfaces are unknown and difficult to calculate when they undergo topological changes. The phase field approach has been shown to be a powerful tool for modeling such problems, considering a known and fixed computational domain. In this context, the Cahn-Hilliard equation, initially developed to model the separation of binary alloys, has been widely used in several applications ranging from modeling tumor growth to image processing. It consists of a non-linear fourth-order parabolic partial differential equation that presents great challenges in the numerical solution, which in certain cases can present non-physical oscillations and demand the use of extremely refined meshes and time steps. This work aims to overcome such numerical difficulties through hybrid finite element formulations in space and second-order formulations in time aiming at robustness and efficiency. The classical Cahn-Hilliard equation as well as other models based on it are studied from a numerical point of view to verify the order of convergence of the presented methods and evaluate their efficiency and precision. In particular, some applications of the Cahn-Hilliard equation such as in the modeling of tumor growth and the electrowetting process are also considered in this work.