An Adaptive Discontinuous Finite Volume Element Method For The Allen-Cahn Equation And Cahn-Hilliard Equation

The Allen-Cahn equation and Cahn-Hilliard equation are the most basic equations describing the phase field models,which can be used to describe the diffusion properties of two-phase interface.They have broad application prospects in materials science,fluid mechanics and engineering problems,and have rich physical background and far-reaching research value.The main difficulties of the Allen-Cahn equation and the CahnHilliard equation in numerical simulation are the existence of nonlinear terms and the strong rigidity caused by small parameters.Therefore,designing efficient and accurate numerical solutions is very important.The main purpose of this article is to use the discontinuous finite volume element method to solve the Allen-Cahn equation and Cahn-Hilliard equation.The arrangement of the entire text is as follows:In chapter 3,we use the discontinuous finite volume element method to solve the Allen-Cahn equation,in which the space is discretized by discontinuous finite volume element method,and the corresponding semidiscrete scheme is obtained.The time is discretized by a fully implicit scheme,and a fully discrete scheme for solving the Allen-Cahn equation with discontinuous finite volume elements is obtained.Meanwhile,it is proved that the fully discrete scheme is conditional energy stable,and the existence and uniqueness of the solution is proved by theoretical analysis.At last,the errors of semi-discrete scheme and full discrete scheme are estimated.In order to verify the effectiveness of this method,we give a series of numerical experiments,in this process,we use the adaptive mesh technology,which can accurately guide the mesh adaptive encryption near the interface,and the obtained numerical experimental results verify the correctness of the theoretical derivation.In chapter 4,we solve the Cahn-Hilliard equation by using the discontinuous finite volume element method combined with the convex split scheme.We prove the important theoretical results of mass conservation and energy dissipation in the fully discrete scheme of the Cahn-Hilliard equation,and then give the error estimation of the semidiscrete scheme of the Cahn-Hilliard equation.The effectiveness of the proposed method is verified by numerical experiments.Finally,a summary of the research content and achievements was made,and prospects for future work were made.