FEniCSx-Based Effective Scheme for the Cahn-Hilliard Equation

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is challenging and time-intensive to solve using conventional methods. Although the equation is difficult to solve, it finds its use cases in many aspects of science including but not limited to the study of phase separation in the binary, isotropic and isothermal mixtures of materials such as molten alloys. Some of the other applications are Bioinformatics, Computational Biology, Biomedical informatics, human activity recognition, Healthcare, computer vision, Climate Science and Natural Language Processing. Many researchers have come up with a way to deal with partial differential equations, but very few works and methods deal with different orders of partial differential equations. In this article, a novel and effective scheme for solving the Cahn-Hilliard equation is proposed using FEniCS. The time of execution and relative tolerance of the solutions of this equation are computed from methods of direct computation and parallel processing which is carried out for the equation with higher-order finite element method for various boundary conditions to demonstrate the efficiency of the proposed approach. The solution thus obtained helps in understanding the non-linear behaviour of the equation with the help of a three-dimensional heatmap plot. This work solves the Cahn Hilliard equation by using the in-built libraries, Dolphin an open-source software called FEniCS and Message Passing Interface (MPI). The MPI is responsible for achieving parallelism in high-performance computing environments and breaks down the problem into four smaller parts making it faster to get to the solution. The obtained results demonstrate that this approach outperforms the existing literature by achieving a forty percent faster computational speed. The approach will be very useful in solving many similar problems arising in the field of science and engineering.