Analysis Of The Improved Element-Free Galerkin Method For Nonlinear Poisson-Boltzmann Equation

With the development of modern science and technology and the general application of computers,numerical calculation plays an increasingly important role in scientific research and engineering analysis.The nonlinear Poisson-Boltzmann equation comes from the fields of biology,physics and chemistry.Researchers hope to find an efficient numerical method to solve this kind of equation.Meshless method is one of the hot research topics in computing science.Element-free Galerkin method is a meshless method which has been researched deeply.At present,we have not found a meshless method to study the nonlinear Poisson-Boltzmann equation.Based on the improved moving least square approximation,the nonlinear Poisson-Boltzmann equation is solved and analyzed by the improved element-free Galerkin method.The first chapter introduces the research background of nonlinear Poisson-Boltzmann equation,traditional numerical calculation method and element-free Galerkin method.In the second chapter,the mathematical theories of the moving least square approximation and the improved moving least square approximation are introduced.And the expressions of approximate function and shape function are given.In chapter 3,the nonlinear Poisson-Boltzmann equation is directly solved and analyzed by the improved element-free Galerkin method.Combining the improved moving least square approximation with Galerkin weak form,we obtain the nonlinear algebraic equations which can be solved by the MATLAB subroutine fslove.And the improved element-free Galerkin method is directly establish for the nonlinear Poisson-Boltzmann equation.Based on the error results of the improved moving least square approximation,the error of the improved element-free Galerkin method for nonlinear Poisson-Boltzmann equation is derived theoretically.Error estimation is obtained in the Sobolev space.Numerical examples verify the theoretical analysis.In chapter 4,the nonlinear Poisson-Boltzmann equation is indirectly solved and analyzed by the improved element-free Galerkin method.Combining the improved moving least square approximation with linearized the Poisson-Boltzmann equation Galerkin weak form,the improved element-free Galerkin method is indirectly establish for the nonlinear Poisson-Boltzmann equation.Error estimation is obtained in the Sobolev space.Numerical examples verify the theoretical analysis.The last chapter gives some conclusions and prospects.