An Indirect Boundary Node Method For Three Elliptic Boundary Value Problems

Laplace equations,Helmholtz equations,and modified Helmholtz equations arise in the fields of mathematics,physics,and engineering.Finite difference method,finite element method and boundary element method are used to solve the boundary value problem of these three elliptic equations.These numerical methods need to be divided into grids,which will bring a lot of inconveniences.The boundary-type meshless method can reduce the dimensions of the required problem and does not require meshing.The boundary point method is a more mature boundary-type meshless method.Based on the indirect boundary integral equation,the meshless indirect boundary point method for solving these three elliptic boundary value problems is given.The first chapter introduces the research background of these three kinds of elliptic boundary value problems firstly,then introduces the research status of meshless,then introduces the mobile least squares approximation theory,and the second chapter introduces the three-dimensional based on indirect boundary integral equations.Boundary Point Method for Dirichlet Problem of Laplace Equation.First,the three-dimensional Laplace equation is transformed into an indirect boundary integral equation by the monolayer potential theory.Then the boundary point method is used to discretize the indirect boundary integral equation.Finally,a numerical example of the three-dimensional Laplace equation is given.In Chapter 3,the boundary point method for the Dirichlet problem of the three-dimensional Helmholtz equation based on the indirect boundary integral equation is introduced.First,the three-dimensional Helmholtz equation is transformed into an indirect boundary integral equation by the monolayer potential theory,then the boundary point method is used to discretize the indirect boundary integral equation.Finally,a numerical example of the three-dimensional Helmholtz equation is given.In chapter four,the boundary point method of the Dirichlet problem for the three-dimensional modified Helmholtz equation based on the indirect boundary integral equation is introduced.Firstly,the three-dimensional modified Helmholtz equation is transformed into an indirect boundary integral equation using the monolayer potential theory,and then the boundary point method is used to discretize the indirect boundary integral.The equation gives a numerical example of a three-dimensional modified Helmholtz equation.Three-dimensional Helmholtz,three-dimensional modified Helmholtz will have weak singularity when calculating integrals with basic solutions.In Section IV,Chapter IV,the calculation of weak singular integrals is given in detail.Numerical example shows that the proposed indirect boundary point method to solve the three kind of elliptic boundary value problems is very effective,compared with the traditional numerical method,the indirect boundary point method is of higher precision,faster convergence speed.