New Types Of Boundary Integral Equations For Several Familiar Stationary Boundary Value Problems And Numerical Methods

In this thesis, the main research includes three aspects. They are boundary integral-differential equation and boundary element method for Neumann boundary value problem of Helmholtz equation in three dimensions, a boundary element method for Signorini problem in three dimensions, and the new types of boundary integral equations for partial differential equations such as stationary convection-diffusion equation in three dimensions. Firstly, the solution of Neumann boundary value problem of three-dimensional Helmholtz equation will lead to solve hyper singular integral equation, when double layer potential distribution formulation is used. In the present work, the relation between normal derivative and tagential derivatives is derived, and the partial derivative of singular kernel is shifted to the unknown function in the integral equation by performing integration by parts. So the new type of boundary integral–differential equation is gotten. Based on this equation, boundary variational formula equivalent to it and Galerkin BEM cooresponded with it are established, and the error estimates in the sense of various norms are analyzed. Secondly, a Signorini problem in three dimensions is reduced to a variational inequality on the boundary, and the existence and uniqueness are established. Furthermore, error estimates for the approximate solution of Signorini problem are given. Finally, starting from Green formulas, single field conservation integral formulas and double field conservation integral formulas have been established for several familiar partial differential equations in three dimensions. By means of these formulas, several new types of boundary integral equations are derived, and the solutions are gotten by double-equation method.