Boundary Element Methods And Their Extrapolations For The Discontinuous Media Problem Based On The Domain Decomposition

This thesis firstly studies high accurate mechanical quadrature methods(MQM) and midpoint constant element collocation methods(MCECM) and their extrapolations for solving boundary integral equations of the discontinuous media problemâ–½.(γ(x)â–½u(x))=0.The author first discusses mechanical quadrature methods and their Richardson extrapolations for the case which the boundary and interfaces are all smooth. The differential equation of the discontinuous media problem is converted into equivalent Fredholm boundary integral equations of the second with logarithmic weakly singularity by using the theory of single layer potential and domain decomposition. A class of mechanical quadrature methods are developed for solving the singular boundary integral equations based on Sidi-Israeli's rule, which possesses a high order accuracy O(h³), less computational complexity and asymptotic expansion of the errors. By the theory of collectively compact and asymptotically compact convergence, and combining with the Euler-Maclaurin expansion, the convergence and stability of approximation solutions are proved theoretically. We also prove that the errors have more than O(h³) order asymptotic expansions. By means of Richardson-h³ extrapolation, an approximation with a higher accuracy order O(h⁵) is obtained. Moreover a posteriori error estimate for the algorithm is derived, which can be used to constructed adaptive algorithm. Numerical examples show that our computing results are good agreement with the theoretical estimations for both extrapolated and non-extrapolated rates of convergence. The cost of CPU time of MQM is much less than that of midpoint constant element collocation method(MCECM).The next, we research the mechanical quadrature methods and their splitting extrapolations for solving the boundary integral equations of the discontinuous media problem with polygonal boundary and interfaces. Because the solutions of the boundary integral equations have singularity at corners, they are not suitable to be discretized directly. So we first use sin^m periodical transformation to eliminate the singularity of the solutions at comers, and then use the Sidi—Israeli's rule and midpoint trapezoidal formula to construct MQM of boundary integral equations of the discontinuous media on polygonal regions. The convergence and stability are also proved theoretically, multi-parameter asymptotic expansions of errors with more than h_ij³(i=1, 2, j=1, 2, d_i) power are derived for the algorithm. Using multi-parameter asymptotic expansions, splitting extrapolation with high accuracy order O(h⁴) are proposed, which can be reduce computing time by parallel computation.The third, a modified constant collocation method is proposed for the discontinuous media problem with polygonal domain. Before the boundary integral equations are discretized, we first transform every boundary by sin^m periodical transformation, then take mid-points of subintervals as the collocation points, at last we get a new mid-point constant collocation method. Numerical results show that the approximation order of the modified mid-point constant method for polygonal domain reaches the same level as the standard mid-point constant method for smooth boundary: i.e., For the second kind boundary integral equation, the approximation order is O(h²); and for the first kind boundary integral equation, the approximation order is O(h³). Since collocation methods are used widely by engineers, this study has important theoretical and practical value.Fourthly, we propose the mechanical quadrature methods for the direct boundary integral equations of the discontinuous media problem with polygonal boundary and interfaces based on the basic solution. Numerical computing show that the MQM for the BIE based on the single layer potential presented before are not good for the case which the discontinuous medias have no inclusion relation(See the right of figure 5.1of chapter 5). The MQM based on the direct boundary integral equations of domain decomposition have extensive adaptability, and the accurate order of approximation solution also reaches O(h³). But since the methods require to calculate two unknows u(x) and (?)u(x)/(?)n on the interfaces while the MQM based on the single layer potential only require to calculate one unknow z₂(σ), their computation amounts are much more than that of the MQM based on the single layer potential. However, as the unknows of the direct boundary integral equations on the boundary have physical meaning, the methods can be easily accepted by the engineers.The fifth, a high accuracy modified collocation method is proposed for the firstkind boundary integral equations of Laplace problem on polygonal regions. If midpoint constant element collocation methods are directly applied to the problem, the accuracy order of the approximation solution can only achieve O(h^β+3/2) at interior point, whereβ=(1-α)/α,απis max internal angle. So the accuracy order is lower than O(h²) at interior point for the concave polygon. To improve the accuracy of the approximation solution, we first transform every edge of the polygonal regions with a suitable sin^m periodical variable transformation before equations being discretized, and next take the mid-points of subintervals as collocation points of boundary integral equations, finally we get a new midpoint constant element collocation method. The method not only adapt to the interior problem, but also adapt to the exterior problem. Numerical results indicate that the accuracy order of the approximation solution can reach O(ha) both for convex polygon and concave polygon at interior point.At last, the thesis use domain decomposition algorithm for the partial differential equation to study binomial deliverability equation of partial penetrating gas well in bottom water reservoir. The drainage domain of partial penetrating gas well in bottom water reservoir consists of the following two parts: (1) Non-Darcy binomial radial flow near the well; (2) Three dimensional flow far away from the partial penetrating gas well due to bottom water drive. Because the main drive mechanism is bottom water drive, the effects of lateral boundaries are negligible, without losing generality, in this paper, we assume the gas reservoir is infinite with impermeable upper boundary and constant pressure lower boundary. This paper presents pressure draw-down equation of partial penetrating gas well in Darcy-flow dominated domain in bottom water reservoir, and combining Non-Darcy binomial radial flow equation, then binomial deliverability equation is obtained. In the literature, the flow equations for partial penetrating wells are based on two-dimensional model with impermeable upper and lower boundaries, those equations can not reflect the characteristics of bottom water drive mechanism, so the new equations in this paper can account for the characteristics of Non-Darcy flow of partial penetrating gas wells in bottom water reservoir with high accuracy.