Research On The Coupling Algorithm Of Moving Mesh And Finite Element Stabilization Method

The convection-diffusion-reaction equation is applied in many practical projects,and its solutions often exhibit phenomena such as large gradients and discontinuities.There are many methods for solving convection-dominated convection-diffusion-reaction equations,and the finite element method is currently a widely used numerical method,but its numerical results still exhibit false oscillations or instability.At the same time,the computational efficiency of the finite element method in solving such problems still needs to be improved,especially in solving some strong convection-dominated problems.Therefore,this dissertation adopts the moving mesh method and stabilization method to study the convection-dominated convection-diffusion-reaction equation and proposes coupling algorithms that can effectively solve the convection-dominated convection-diffusion-reaction equation and obtain a relatively ideal numerical result.These algorithms are the moving mesh variational multi-scale finite element method(MM-VMFEM)and the moving mesh streamline upwind Petrov-Galerkin method(MM-SUPG),respectively.The full text is divided into six chapters,and the content of each chapter is summarized as follows:Chapter 1 introduces the research background and significance of this topic,as well as the current research status of the moving mesh method and its coupling algorithm with variational multiscale methods(VMS)and streamline upwind Petrov-Galerkin methods(SUPG).The main research content of this paper is elaborated.Chapter 2 introduces some basic knowledge,including the basic format of finite element methods,moving mesh methods,variational multiscale methods,and the basic principles of the SUPG.In Chapter 3,the moving finite element method(MFEM)proposed by Li et al.is applied to solve three Burgers’ equations with high Reynolds numbers,and stable numerical solutions are obtained,which shows the effectiveness of the MFEM in solving such problems.In Chapter 4,based on the stable numerical solutions obtained by the MFEM,the moving mesh method is coupled with the variational multiscale finite element method to propose the MM-VMFEM.Numerical experiments are conducted on three class convectiondominated convection-diffusion-reaction equations,and the numerical results show that even when dealing with strong convection-dominated convection-diffusion-reaction equations,the MM-VMFEM can still obtain stable numerical results.Chapter 5 extends the coupling algorithm of the moving mesh method and the stabilization method to solve the magnetohydrodynamic(MHD)flow equation.The decoupled MHD flow equation is a typical convection-diffusion-reaction equation,and high Hartmann number cases correspond to convection-dominated cases.We couple the moving mesh method with the SUPG and propose the MM-SUPG to solve a classical MHD flow problem.We discuss the transverse and oblique magnetic fields separately.The numerical results show that to some extent,the MM-SUPG can improve the stability of the finite element method.Up to now,we have obtained ideal contour results for magnetic and velocity fields.At the same time,we also provide numerical solution images for reference,indicating that the MM-SUPG can obtain more effective numerical solutions when solving high Hartmann number MHD flow equations.Chapter 6 is a summary of the entire text,indicating the main innovative points presented in the paper and providing directions for further research for future research.