Theory Of Element-free Galerkin Method And Its Application In Flow Problems

The meshless(or meshfree)method,which relies on scattered nodes rather than meshes,is a class of newly developing numerical methods in recent years.It is one of the hotspots of current numerical methods.As we all know,mathematical theory of the meshless method is incomplete,which more or less restricts its the development and application.The current thesis is dedicatede to the theoretical analyses and numerical applications of the element-free Galerkin method for solving the second-order elliptic mixed boundary value problem and the incompressible flow problems.The main contents of this thesis are as follows:Firstly,priori approximate estimatations of the element-free Galerkin method for the second-order elliptic mixed boundary value problem are researched.By using the penalty method to apply the Dirichlet boundary condition,the existence and uniqueness of the solution to the Galerkin variational problem for the penalized second-order elliptic mixed boundary value problem are strictly proved.Based on the error estimations of moving least-squares approximation in Sobolev space,theH~1 and ~2L error estimations of the element-free Galerkin method for second-order elliptic mixed boundary value problems are studied.The error results show that theH~1 and ~2L error estimations of unknown variable are related to the selection of the basis function,the node spacing and the penalty factor.Secondly,priori approximate estimatations of the Stokes problem and the penalized Stokes problem are researched.The existence and uniqueness of the solution to the Galerkin variational problem for the penalized Stokes problem are proved.Meanwhile,the existence and uniqueness of discrete solution to the nonstandard element-free Galerkin method for the penalized Stokes problem are proved.Similarly,by means of the error estimations of moving least-squares approximation,the error estimations of velocity and pressure are analyzed.The error results show that the error estimations of velocity and pressure are related to the selection of the basis function,the node spacing and the penalty factor.Then,based on the basic idea of generalized finite element method(GFEM),the generalized element-free Galerkin(GEFG)method is developed to solve the stationary Stokes problem.The comparative analysis shows that GEFG is similar to the variational multiscale element-free Galerkin(VMEFG)method in the variational multiscale framework,but the former is more reasonable in practical problems and its discrete form is simpler and more direct.Numerical experiments show that it has high computational efficiency and accuracy.Finally,the variational multiscale interpolating element-free Galerkin(VMIEFG)method is developed to solve the Darcy-Forchheimer model and the generalized Oseen problem.In the proposed method,the interpolating moving least-squares method and the moving Kriging interpolation(MKI)are selected to construct the meshless shape functions,respectively.The present method is based on a decomposition of velocity and weighting function of the velocity into coarse and fine scales.By solving the fine scale problem analytically,stabilization parameters can appear naturally.The VMIEFG method allows equal-order basis for velocity and pressure,i.e.,the standard element-free Galerkin method is available,so programming is easy to implement.Numerical experiments show that the proposed method has good stability and numerical accuracy.