Analysis Of The Element-free Galerkin Method For A Class Of Linear Or Nonlinear Differential Equations

Many linear or nonlinear differential equations come from the mathematical models of many problems in natural science.Researchers hope to find efficient numerical methods to solve these equations.Meshless method is one of the hot research topics in computing science,in which the element-free Galerkin method is one of the most usually applied methods.In this dissertation,the element-free Galerkin method is used to analyze the linear transient heat conduction equation and the nonlinear Burgers equation.The details are as follows.In the first chapter,the research background of several traditional numerical methods,meshless method,and the element-free Galerkin method is introduced.In the second chapter,the basic principles of the moving least squares approximation for constructing meshless shape functions and the second order backward differential formula for time discretization are described.In the third chapter,the element-free Galerkin method is presented to solve the transient heat conduction equation with mixed boundary conditions.The second order backward differential formula is applied to approximate the time derivative terms of the given equations.Combining the Galerkin weak form with the moving least squares approximation,the element-free Galerkin method for the transient heat conduction problem is established.Then,based on the error results of the moving least squares approximation,the theoretical error of the element-free Galerkin method is gained in the Sobolev space.Finally,the efficiency of the element-free Galerkin method and the theoretical analysis are verified by several numerical examples.In the fourth chapter,the nonlinear Burgers equation is analyzed and solved by the element-free Galerkin method.The second order backward differential formula is used to approximate the time derivative terms of the nonlinear Burgers equations.The nonlinear terms of two dimensional viscous Burgers equation are linearized.Then,combining the Galerkin weak form with the moving least squares approximation,the element-free Galerkin method for the nonlinear Burgers equation is established.Finally,the efficiency of the element-free Galerkin method for the nonlinear Burgers equation is verified by several numerical examples.In the fifth chapter,the dissertation summarizes and looks forward to the research work.