Study Of Numerical Methods For Surface Partial Differential Equations

Surface partial differential equations models have important theoretical significance and practical value in many research areas such as materials science,biology and computer graphics.Research on the related model theory and numerical methods is the front of computational physics and biology,and has attracted wide attention at home and abroad over the past decade.In practical application simulation,due to the geometric complexity of the surface,it is very difficult to solve the surface partial differential equations by analytical approaches.Therefore,it is very important to develop accurate,stable and efficient numerical methods.This thesis is devoted to the improvements of the finite element method for the parabolic type equations and convection-dominated diffusion problems on surfaces,and the study of the local tangential lifting method for the diffusion-reaction systems,the problems with singular sources and the moving interface problem on surfaces.The main specific research contents and results of the thesis are as follows:Firstly,the lumped mass finite element method for the surface parabolic equation is developed.The surface parabolic equation satisfies the maximum principle.To construct a numerical method that preserves the discrete maximum principle,the multi-dimensional lumped mass method is extended to the finite element method on surfaces.The lumped finite element method is an improvement of the standard linear finite element method which lacks discrete maximum principle preservation,and is also applicable to the linear surface finite element space.The idea of the method is to make a diagonal lumping modification on the finite element mass matrix so as to make the coefficient matrix of the algebraic equations to be the M-matrix,which enables scheme to be maximum principle preserving.The related error analysis and the proof of discrete maximum principle preservation are provided.The validity of the method is also verified by numerical experiments.Secondly,several discrete maximum principle preserving schemes are presented to solve the surface Allen-Cahn type phase field equations.The surface Allen-Cahn type equations are nonlinear parabolic equations often used to simulate multiphase flows on surfaces.They achieve the effect of tracking the interface of mixtures by describing the variation of the concentrations of the mixtures' components.Using the finite element method to solve the surface Allen-Cahn equations which have small free energy parameters often leads to numerical oscillations and may be not maximum principle preserving.In order to obtain stable and high-resolution numerical solutions,the stabilized semi-implicit,convex splitting and operator splitting schemes combined with the lumped mass finite element method are proposed to solve the standard,conservative Allen-Cahn equations on surfaces,and are proved to be maximum principle preserving theoretically.The proposed numerical schemes are used to simulate the phase separation phenomena and the mean curvature flows on surfaces.The results confirm the reliability and discrete maximum principle preservation of the proposed schemes.Thirdly,for the stationary convection-dominated diffusion problems on surfaces,the well-posedness and error estimates of the finite element method is established.According to the estimations and computations,it can be confirmed that solving the convection-dominated diffusion problems on surfaces by the standard finite element method may lead to strong numerical oscillations.In order to improve the stability,the streamline diffusion stabilization method is extended on surface.To further improve the computational efficiency,i.e.,achieving a more accurate and higher resolution solution with relatively few degree of freedoms,a gradient recovery-based adaptive streamline diffusion finite element method is provided.To show the stability and the efficiency of the proposed method,a series of numerical examples are illustrated.Fourthly,there is still an instability issue of the standard finite element method for unsteady convection-dominated diffusion problems on surfaces.To overcome this drawback,this thesis extends the classical characteristic finite element method to enable it can be performed on surfaces.By combining with the previous lumped mass method,a positive preserving characteristic finite element method is presented.As an application of the proposed method,the characteristic finite element method combined with a decoupling method is employed to solve the surface chemotaxis models which describes the aggregation of biomes.The relevant theoretical analysis of the positivity preservation is shown.A large amount of numerical examples are performed to not only show the effectiveness of the methods,but also to make a series of numerical explorations on the convection-dominated diffusion heat and mass transfer phenomena and chemotaxis phenomena on surfaces.At last,the local tangential lifting method is an ideological method for constructing discrete derivatives or functions on surfaces.This method extends the surface functions to the local tangent planes of surfaces,then constructs discrete derivatives or functions on tangent planes as approximations of derivatives or functions of original surfaces.It simplifies the discretization problem on local surface as the discretization problem on local two-dimensional domain and is easy to be programmed.It is also a basic idea to construct meshless method for solving partial differential equations on surfaces,which avoids the global surface triangular mesh used in the finite element method.Based on this method,a discrete scheme of the Laplace-Beltrami operator is presented and applied to solve the diffusion-reaction systems.And a discrete delta function method is presented to solve the problems with singular sources on surfaces.Moreover,the above two proposed methods are combined with a front tracking method to solve a moving interface problem on surfaces.Various of numerical examples are shown to demonstrate the effectiveness of the proposed methods.