An Improved Element-free Galerkin Method For Helmholtz Equation

After several decades of developments,there are a variety of meshless methods in all of numerical computation.At present,many scholars at home and abroad combine moving least-square approximation with the approximating function of meshless method to form an important method——element-free Galerkin method(EFG).This method has been widely used in engineering,but its corresponding mathematical theory is not perfect.In order to better promote its application,this paper discuss with improved algorithm to solve the boundary value problem of Helmholtz equation and modified Helmholtz equation in detail.The first chapter describe several kinds of traditional numerical method,the research background of moving least-square approximation(MLS),the improved moving least-square approximation(IMLS),Helmholtz equation and modified Helmholtz equation and EFG.The second chapter detailed introduce the theory of MLS and IMLS.On the basis of the IMLS,we discuss the stability of the IMLS and construction of shifted and scaled basis function.Through the analysis of stabilied IMLS,the method has better stability and accuracy.The third and fourth chapter is the major work,which we combines stabilied IMLS with Galerkin weak integral form of Helmholtz equation and modified Helmholtz equation to establish Neumann,Dirichlet and the mixed boundary value problems of improvement EFG.At the same time,we prove that the theoretical error estimates of modified Helmholtz equation boundary value problem in the Sobolev space.Finally,numerical examples are given to verify the theoretical error results.