Virtual Element Method For Linear Convection-diffusion Problems

Convection-diffusion problem is one of the most basic models in fluid mechanics.It is mainly used to model the variation of a physical quantity carried by a fluid particle,such as temperature or the concentration of substances dissolved during the process of fluid.The problem has wide application in the field of environmental science,fluid mechanics and electronic science.In practical numerical simulation,fluid mechanics calculation is usually modeled by the Lagrangian method to ensure the precise description of the material interface,however,the fluid flow usually causes a large deformation of the Lagrangian grid,resulting in a decrease in the accuracy of many computational schemes applied in the orthogonal grid,and no convergence.To avoid grid re-partition and physical quantity re-mapping which will increase the computational complexity and numerical dissipation,it is very important to construct a high-precision calculation formulation for solving convection-diffusion problems directly on arbitrary polygons,non-smooth and severely distorted grids.This dissertation focuses on designing numerical methods on arbitrary polygons,non-smooth and distorted grids to achieve efficient solution.The contents and results of this dissertation are as follows:Based on the virtual element method(VEM),we use the lowest order virtual el-ement discretization to solve the convection-diffusion equation.Firstly,we study the tirme-dependent diffusion-convection-reaction equation.In time,we use the Crank-Nicolson scheme,and in space,we use VEM to construct the discrete scheme.The error estimates of semi-discrete and full-discrete schemes are given.The optimal convergence order of the discrete scheme in time and space is obtained.The numer-ical experiments show the effectiveness and feasibility of the method.Secondly,we study the convection-dominated diffusion equation.When solving the convection-dominated diffusion equation directly with VEM,numerical oscillation often occurs.To deal with the numerical oscillation caused by the large Peclet number,this dissertation draws on the traditional numerical method and stabilization method,and proposes a kind of stabilized VEM,which is a combination of virtual element method and streamline diffusion method(SD-VEM).The main characteristic of this method is that the trial function in the stability term comes from the self-adjoint op-erator-?·(K(x)?v)-b(x)·?v.The stabilization method takes into account both convection effect and diffusion effect,and the selection of stabilization parameters is relatively simple.By introducing the energy norm,we prove the optimal conver-gence order of the stabilization method.A series of benchmark tests are presented to show that the SD-VEM is valid for convection-dominated diffusion problem and can better simulate boundary layer problems.