| Nonlinear partial differential equations have applications in many fields,such as industrial manufacturing,weather forecasting,reservoir simulation and new energy source development.Because the understanding of Natural essence is limited,and it is very difficult to solve the problem,the numerical simulation is an important tool to understand the change law of the nonlinear problem,but the direct numerical simulation of nonlinear partial differential equation has the following difficulties: large computing scale,long time,nonlinear and coupling between the variables.Therefore,it is very important to find the algorithm with long time stability,high efficiency and low consumption.The projection method is an efficient numerical format for dealing with multivariable coupling models.The algorithm decomposed the original model into a number of small linear problems which reduces the size of the solution,achieve the goals of the small computation and saving time.This paper focuses on constructing the projection method for the Kelvin-Voigt model:In chapter 3,we consider the first-order projection format of the Kelvin-Voigt model.By introducing intermediate variables,the original problem is decomposes into two linear subproblems.The energy method and negative norm technique are used to establish the unconditional stability and convergence of numerical solutions.Then numerical examples are given to verify the theoretical results.In chapter 4,the characteristic projection finite element method is introduced to deal with the Kelvin-Voigt model,which implicitly deals with the nonlinear terms,gives the stability of the numerical solutions,and obtains the optimal error estimation.Finally,numerical examples are given to verify the theoretical results.In chapter 5,we consider the second-order Gauge-Uzawa algorithm which uses the first-order projection format numerical solution,and we obtain the stability analysis and convergence theorems of the numerical solution of Kelvin-Voigt model. |
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