High Accuracy Algorithms For Solving The Indirect Boundary Integral Equations Of Steady State Heat Conduction Problems

Integral equations have been used in many science and engineering problems. Inthis dissertation, indirect boundary element methods are systematically applied to studythe numerical solutions of the boundary integral equations of steady state heat conductionproblems with Dirichlet conditions. For the steady state anisotropic heat conduction prob-lems, we discussed the smooth domains and concave polygons by mechanical quadraturemethods, respectively. For the isotropic case, we study the three dimensional axisymmet-ric Laplace problems by modified quadrature method. These work are stated in detail asfollows.High accuracy mechanical quadrature methods are applied to solve boundary integralequations of anisotropic Darcy’s equations with Dirichlet conditions on smooth domains.By using single potential theory and fundamental solution of Darcy’s equations, Darcy’sequations can be converted into the first kind integral equation with a logarithmic and sin-gular kernel. Combining with middle point rule，the mechanical quadrature methods areconstructed for solving the weakly singular boundary integral equation system of the firstkind. By estimating the sup-bound and inf-bound of the eigenvalue expression of discretematrix, the condition number of discrete matrix is obtained only O(h1), which showthe mechanical quadrature methods possess the excellent stability. The convergence andstability are proved based on Anselone’s collective compact and asymptotical compacttheory. Furthermore, an asymptotical expansion with odd powers of errors for mechani-cal quadrature methods is presented, which possesses high accuracy order O(h3). Usingh3Richardson extrapolation algorithms, the accuracy order of the approximation can begreatly improved to O(h5), and an a posteriori error estimate can be achieved for con-structing self-adaptive algorithms.Numerical solutions for boundary integral equations of anisotropic Darcy’s equationswith Dirichlet conditions on polygonal boundaries are studied by mechanical quadraturemethods. In concave polygons, the solutions at concave points have singularities. Itsgreatly dampens the approximate accuracy. Hence, it has been critical point for math-ematicians to overcome the difficulty for a long time. The accuracy of Galerkin finitemethod is only O(h1+r)(r <1) and the accuracy of collocation methods are even lower. The logarithmic singularities at the corner points of the boundary can be removed bysin transformation. Then the discretion matrix by mechanical quadrature methods andthe asymptotic expansions of errors are obtained, which show the convergence order isO(h3). The convergence rate O(h5) can be achieved after using the splitting extrapola-tion methods.The fully anisotropic heat conduction problems with Dirichlet boundary conditionsby mechanical quadrature methods are investigated on smooth domains and polygonsrespectively. The convergence of mechanical quadrature methods are proved by Usingcollectively compact theory. Increasing the boundary nodes, the effect and stability willbe well.Modified quadrature method for three dimensional steady state isotropic heat con-duction problems are studied. By indirect boundary element method, the axisymmetricLaplace problems can be converted into the single boundary integral equations of the firstkind. Then, the periodic transformation can be used to remove the singularities of the cor-ners or endpoints. This technique improves the accuracy of modified quadrature method.The modified quadrature formula is simple to work for the singular integrals. Numericalexamples show the convergence rate of the modified quadrature method is O(h3).