On Spurious Oscillations At Layers Diminishing Methods For Convection-diffusion Models On Surface

Surface partial differential equations(PDEs)are posed on differential manifolds em-bedded in three-dimensional region,which occur in many applications.As a kind of important PDEs,convection-diffusion equations are naturally used in the establishment of mathematical models posed on surfaces.The difficulties of solving the convection-diffusion equations on surfaces are the same as that on plane,that is,when transport processes on surfaces dominates over diffusion,standard FEMs,FDMs and FVMs tend to be unstable unless the mesh is sufficiently fineIn this thesis,two types of spurious oscillation at layers diminishing(SOLD)meth-ods are extended to solve the convection-diffusion equations posed on surfaces.One is edge stabilization method based on continuous internal penalty,which can preclude spu-rious oscillations along the sharp layers by adding penalty terms about the gradient of the solution,the stability and convergence results of the method are given.The other is called Mazukami-Hughes method that is a Petrov-Galerkin method satisfying the discrete maximum principle by choosing an appropriate convection vector b in the direction per-pendicular to ?u.In addition,we also consider a kind of chemotaxis models to describe the biological aggregation behavior on surfaces.We successfully simulate the chemotaxis model by combining Mazukami-Hughes method,mass correction method and surface operator recovery technique,the corresponding analysis of positivity preservation is also given.