High Accuracy Algorithms For Solving Three Dimensional Axisymmetric Boundary Integral Equation

Many mathematical models in science and engineering can be described as the boundary value problems of differential equations in three dimensional domains of revolution,which are called axisymmetric boundary value problems,and are the hot spot in the current research.In this dissertation,the boundary element method is used to transform these problems into axisymmetric boundary integral equations,and the mechanical quadrature method is applied to solve the axisymmetric elastostatic boundary integral equations,the axisymmetric Darcy's boundary integral equations,the axisymmetric nonlinear Laplace's boundary integral equations and the axisymmetric Poisson's boundary integral equations.These work are stated in the follows.High accuracy quadrature methods are applied to solve axisymmetric elastostatic equations with Dirichlet conditions.Based on the single potential theory and the fundamental solution of axisymmetric elastostatic equations,the axisymmetric elastostatic equations can be converted into the first kind boundary integral equations with logarithmic weakly singular kernel.since the rotation plane boundary is not smooth,the solutions of integral equations at the corner are non smooth,even singularity,By using Sidi transformation,the singularity at the corner can be eliminated.By using the Sidi quadrature rules and the middle point rule,the mechanical quadrature method is constructed for solving the weakly singular boundary integral equations of the first kind.By Anselone's collective compact theory,the convergence and stability are proved.Furthermore,the accuracy order (?38)(6)of the approximation can be obtained.Numerical solutions for boundary integral equations of axisymmetric anisotropic Darcy's equations with Dirichlet conditions are studied by the mechanical quadrature method.By using the single potential theory and the space coordinate transformation,the axisymmetric Darcy's equations are converted into the first kind boundary integral equations with logarithmic weakly singular kernel.The mechanical quadrature method is used for solving the boundary integral equations.In order to improve the accuracy of the mechanical quadrature method,the Sidi transformation is used to eliminate the singularity at the corner.The error of the numerical solutions are multi parameter asymptotic expansions with odd order,which indicates the accuracy of the numerical solutions for (?38)(6).Furthermore,the splitting extrapolation algorithm is used to improve the convergence of numerical solutions.the collectively compact theory is used to prove the convergence of the mechanical quadrature method.The boundary integral equations of axisymmmetric Laplace equations with nonlinear boundary conditions are solved by the mechanical quadrature method.By using the direct boundary integral equation method and the fundamental solution of the axisymmetric Laplace's equations,the axisymmetric Laplace equations are converted into nonlinear boundary integral equations with logarithmic weakly singular kernel.the nonlinear approximate equations can be obtained by the mechanical quadrature method and the Newton iteration algorithm.Base on Stepleman theorem the solvability can be proved and the existence of the approximate solutions can be obtained.Moreover,The error of the approximate solutions asymptotic expansions with odd power are obtained,which indicates the accuracy order (?3)of the numerical solutions.The high accuracy order(?5)is obtained by extrapolation algorithm once.The mechanical quadrature method is used to solve the boundary integral equations of axisymmetric Poisson's equations with Dirichlet conditions.By using the particular solutions,the axisymmetric Poisson's equations are deduced from the axisymmetric Laplace's equations.Base on the single potential theory,the equations are converted into the first kind boundary integral equations with logarithmic weakly singular kernel.The Sidi transformation is used to eliminate the singularity at the corner.The mechanical quadrature method is used for solving the boundary integral equations.The error of the numerical solutions are multi parameter asymptotic expansions with odd order,which indicates the accuracy order (?38)(6)of the numerical solutions.By the splitting extrapolation algorithm,the accuracy order (?58)(6)of the numerical solutions are obtained.Several numerical examples are given to verify our theoretical analysis.