| The boundary integral equations (BIE) of the first kind have been widely used in science and engineering and their accuracies are higher than that of BIE of the second kind in the practical computation, but, on the theoretical side, they can not fit within the framework of the classical Fredholm theory. Therefore, few papers have come out to their researches, and only concentrated on Galerkin and collocation methods based on projection theory in the numerical calculation. On the one hand, the mechanical quadrature methods (MQM) for solving BIE of the first kind are void of the collectively compact theory, so they have not been sufficiently analyzed yet; on the other hand, the discrete matrix is full, and generating each element has to calculate an improper integral for the collocation methods or a double improper integral for Galerkin method, however, mechanical quadrature method (MQM) are without calculating any integrals for getting discrete matrix. Hence, it has come to one's notice that the calculation of the discrete matrix becomes very simple and the most of work can be saved. But it has always been a great difficulty in numerical analysis for many years how to construct the quadrature rules and prove their reliability.In this thesis new methods are proposed to deal with difficulties of the practical computation and the above theoretical analyses. Firstly, we make use of Lyness' and Side' quadrature rules to compute weakly singular and singular integral; secondly, by calculating directly we get the eigenvalue expression of discrete matrix in the special case and estimate their sup-bound and inf-bound; finally, by using perturbation theory and Anselons' collective compact theory, we not only obtain that it is reasonable to construct MQM, but also show that thecondition number of discrete matrix is only O(h-1)), which corrects mathematicians' point of view that the numerical solution for BIE of the first kind is not stable. On the basis of the above ways, the thesis first present MQM with the high accuracy for solving BIE of the first kind of Laplace equation, Stokes equation, biharmonic equation, Steklov eigenvalue problem, elasticity equation, nonlinear boundary value problem and three-dimensional axisymmetric problems and so on in scientific and engineering problems on nonsmooth domains. Especially, in concave polygons, since the solution at a concave point has singularity, it greatly dampens the approximate accuracy. Thus, it has been a critical point for mathematicians to enhance the approximate accuracy for a long time. The accuracy of Galerkin methods is only O(h1+ε) (0 <ε < 1) and the accura -cy of collocation methods is also lower than its, while the accuracy of MQM in the thesis is O(h3).It is a very important study field in numerical mathematics how to improve further the approximate accuracy. Over the years, mathematicians always think that the extrapolation of integral equations with non-smooth kernels is no longer founded theoretically. In particular, extrapolation algorithms for solving BIE of the first kind by MQM have no satisfactory result so far. Much less, using the splitting extrapolation method (SEM), a kind of new technique, enhances the approximate accuracy. The thesis uses the specially periodic transform to eliminate singularities of solutions at angle points. Because of independent of mesh parameters on each edge, it is deduced that the errors possess multivariate asymptotic expansion with odd power h2μ+1(μ∈ N). Once some discrete equations about some coarse mesh partitions are solved in parallel, the approximate accuracy will be largely improved by SEM. The thesis sets up SEM for solving BIE of the first kind for the first time. By SEM we can not only get the high accuracy of approximations, but also derived a posteriori error estimate for adaptive algorithms.
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