Superconvergence Methods And Splitting Extrapolations For Solving Three-dimensional Axisymmetric Problems

Many problems in science and engineering can be described as three dimen-sional axisymmetric boundary value problems. This dissertation makes someresearch on the application of the mechanical quadrature methods and splittingextrapolation methods to axisymmetric boundary integral equations. In this dis-sertation we first discuss the feasibility of developing the mechanical quadraturemethods and extrapolation techniques in the axisymmetric Laplace problems,axisymmetric free boundary problems and axisymmetric Stokes problems, theo-retically and methodically.Many engineers have given the boundary element methods their much at-tention, by which the axisymmetric problems can be converted into the singleboundary integral equations of the first kind and second kind. At present, thetheoretical analysis of the axisymmetric boundary integral equations are incom-plete, the numerical computation mainly depends on the Galerkin and collo-cation methods based on projection theory. But the discrete matrices in theGalerkin and collocation methods are full, and the generation of each matrixelement has to calculate an improper integral for the collocation methods or adouble improper integral for the Galerkin methods, which imply that the cost ofcalculating the discrete matrices could be so expensive to greatly exceed that ofsolving the discrete equations. Furthermore, the accuracy of numerical solutionsis also always lower. So, the high computational complexity and low numericalaccuracy limit their extensive application. In this dissertation, we make use of themechanical quadrature methods with Sidi and Lyness quadrature rules to buildthe discrete matrices in axisymmetric boundary integral equations without cal- culating any improper integrals. Hence, the calculation of the discrete matricesbecomes very easy and simple, and numerical accuracy can also be improved.In additional, the fundamental solutions of the axisymmetric problems, dif-ferent from two-dimensional problems, have cauchy singularities on the symmetryaxis. In the Galerkin and collocation methods, an especial treatment will be em-ployed to deal with symmetry axis . On the other hand, the solutions and theirnormal derivatives of the boundary integral equations are usually non-smooth ,even singular, at the corner points if the boundary of axially symmetric body isnon-smooth, and the correlative theoretical investigation becomes very di?cult.In this dissertation, in order to overcome the singular di?culties, the periodictransformations are introduced to eliminate the singularities of solutions at thecorner points and improve the numerical accuracy of the mechanical quadraturemethods to O(h3). At the same time, because of independent of mesh parameterson each segment of the boundary, the multi-parameter asymptotic error expan-sion with power O(h3) can be deduced. Once some discrete equations on thecoarse mesh partitions are solved in parallel, we cannot only further improve thenumerical accuracy by the extrapolation and splitting extrapolation methods,but also give a posterior error estimate. A posterior error estimate means thatour algorithms are adaptive. The research work in this dissertation can be rarelyfound in the past literature about the axisymmetric problems.The work in this dissertation can be divided into three parts.The first is the mechanical quadrature methods and their splitting extrapo-lations for solving boundary integral equations of Laplace problems. The directboundary element method and indirect boundary element method are employedto build the di?erent boundary integral equations. A theoretical investigationshows that error has a multi-parameter asymptotic expansion. The numerical ex-periments show that the numerical accuracy is always lower than O(h3) withoutthe periodical transformations. The numerical results verify our the theoreticalanalysis.The second is the boundary element method of free boundary Issues of sta-tionary water cone for gas wells. The mechanical quadrature methods are firstintroduced into the boundary integral equations of gas well in natural gas indus- try. The spline function is used to simulate the curve of water cone. An iterativealgorithm is designed to obtain the position of water cone. At last, to di?erentpressure at the bottom of well, we calculate the position of water cone. Thenumerical results show that this algorithm is e?ective and objective.The third is Stokes problems. The indirect boundary element method areemployed to build the boundary integral equations of the first kind. The theoret-ical investigation shows that error has a multi-parameter asymptotic expansionwith power O(h3) under the periodic transformations. Practical computationsshow that our method exceeds the Galerkin and collocation methods, whethercomputation complexity or numerical accuracy.