稳态热传导问题的间接边界积分方程的高精度算法
|
摘要
积分方程在科学和工程技术问题中有着广泛应用。本文针对稳态热传导问题,利用间接边界元方法系统地研究了其边界积分方程的数值解。对各向异性热传导问题,采用机械求积法讨论了光滑域和凹角域情形,对各向同性热传导问题,利用校正求积法研究了三维轴对称域情形。在以下几方面展开研究工作，取得创新成果。
1.研究了光滑域上带Dirichlet边界条件的达西方程数值解法。通过单层位势理论,利用达西方程的基本解,把达西方程转化为带对数奇性核的第一类弱奇异边界积分方程,然后借助Lyness与Sidi的弱奇异求积公式,结合中矩形公式,构造了求解含弱奇性核的第一类边界积分方程的机械求积法,直接估计离散方程本征值的上下界,得到离散矩阵的条件数为O(h1),从而表明机械求积法解第一类积分方程具有优秀的数值稳定性。基于Anselone的聚紧收敛理论证明了数值解的存在性和收敛性。此外,证明了近似解的误差有奇次幂的单参数渐近展开,得到机械求积法的收敛阶为O(h3),通过使用h3Richardson外推法,数值精度提高到O(h5)。利用导出的渐近后验误差估计构建了自适应算法。
2.研究了多角域上带Dirichlet边界条件的达西方程数值解法。对于凹角域情形,解在凹点的奇异性将严重地影响近似解的精度,如何提高它的精度,成为数学家们长期关注的热点。Garlerkin有限元的精度一般是O(h1+r)(r <1),而配置法的精度更低。为消除对数奇异和角点奇异,使用了一个三角正弦变换,然后利用机械求积法得到边界积分方程的离散近似方程,并推出了关于误差的含奇数阶的多参数渐近展开式,其表明数值解的精度是O(h3),继而借助分裂外推法消去误差展式中的低阶项得到误差的高阶项,因此提高了机械求积法的收敛阶。
3.研究了稳态完全各向异性热传导方程的数值算法。利用机械求积法,我们分别讨论了光滑域和多角域的情形,借助聚紧理论证明了机械求积法的收敛性。数值算例验证了适度增加边界节点数目，可进一步提高数值解的精度和稳定性。
4.研究了校正求积法解三维轴对称稳态各向同性热传导问题。利用间接边界元,把三维各向同性热传导问题,也即轴对称Laplace问题,转化为一维边界积分方程,然后运用周期变换消除了解在角点处的奇性,进一步提高了校正求积法的精度,该求积方法简单易于实施。数值试验表明该算法的收敛阶为O(h3)。
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).
1.研究了光滑域上带Dirichlet边界条件的达西方程数值解法。通过单层位势理论,利用达西方程的基本解,把达西方程转化为带对数奇性核的第一类弱奇异边界积分方程,然后借助Lyness与Sidi的弱奇异求积公式,结合中矩形公式,构造了求解含弱奇性核的第一类边界积分方程的机械求积法,直接估计离散方程本征值的上下界,得到离散矩阵的条件数为O(h1),从而表明机械求积法解第一类积分方程具有优秀的数值稳定性。基于Anselone的聚紧收敛理论证明了数值解的存在性和收敛性。此外,证明了近似解的误差有奇次幂的单参数渐近展开,得到机械求积法的收敛阶为O(h3),通过使用h3Richardson外推法,数值精度提高到O(h5)。利用导出的渐近后验误差估计构建了自适应算法。
2.研究了多角域上带Dirichlet边界条件的达西方程数值解法。对于凹角域情形,解在凹点的奇异性将严重地影响近似解的精度,如何提高它的精度,成为数学家们长期关注的热点。Garlerkin有限元的精度一般是O(h1+r)(r <1),而配置法的精度更低。为消除对数奇异和角点奇异,使用了一个三角正弦变换,然后利用机械求积法得到边界积分方程的离散近似方程,并推出了关于误差的含奇数阶的多参数渐近展开式,其表明数值解的精度是O(h3),继而借助分裂外推法消去误差展式中的低阶项得到误差的高阶项,因此提高了机械求积法的收敛阶。
3.研究了稳态完全各向异性热传导方程的数值算法。利用机械求积法,我们分别讨论了光滑域和多角域的情形,借助聚紧理论证明了机械求积法的收敛性。数值算例验证了适度增加边界节点数目，可进一步提高数值解的精度和稳定性。
4.研究了校正求积法解三维轴对称稳态各向同性热传导问题。利用间接边界元,把三维各向同性热传导问题,也即轴对称Laplace问题,转化为一维边界积分方程,然后运用周期变换消除了解在角点处的奇性,进一步提高了校正求积法的精度,该求积方法简单易于实施。数值试验表明该算法的收敛阶为O(h3)。
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).
引文
[1]冯康,余德浩.关于调和方程自然积分算子的一个定理[J].计算数学,1994,3:221-226
[2]余德浩.自然边界方法的数学理论[M].北京:科学出版社,1993,1-18
[3]余德洁.无界区域上stokes问题的自然边界元与有限元素祸合法[J].计算数学，1992,14:371-378
[4]余德浩.无界区域非重叠型区域分解算法的离散化及其收敛性[J].计算数学,1996,3:328-336
[5]林群，朱超定.有限元的预处理和后处理理论[M].上海:科学技术出版社，1994,2-13
[6]朱起定，林群.有限元超收敛理论[M].湖南科技出版社，1989,18-28
[7]林群，严宁宁.高效有限元构造与分析[M].河北大学出版社,1996,35-65
[8]祝家麟.定常stokes问题的边界积分方程法[J].计算数学,1986,3:281–289
[9]祝家麟.椭圆边值问题的边界元分析[M].北京:科学出版社，1991,1-37
[10]祝家麟.边界元方法:一个老而新的发展中的数值方法[J].数学的实践与认识,1983,3:44-49
[11]吕涛，黄晋.平面双调和问题的第一类边界积分方程的高精度求积方法与外推[J].应用数学学报,2001,24(3):321-332
[12]吕涛，马长征.第二类边界积分方程nystro¨m解的高精度组合方法[J].计算物理,1994,11:75-84
[13]黄晋,吕涛.曲边多角形区域上第一类边界积分方程的高精度机械求积法与分裂外推[J].计算数学,2004,2:51–60
[14]黄晋,吕涛,申慧蓉.多角形域上Robin问题的边界积分方程的求积法与分裂外推[J].燕京大学学报,2004,2:137-140
[15]吕涛,黄晋.定常Stokes问题的边界积分方程的高精度求积方法与外推[J].计算物理,1999,16:561–567
[16]吕涛,黄晋.第二类若奇异积分方程的高精度Nystrom方法与外推[J].计算物理,1997,14(3):349–355
[17]吕涛,石济民,林振宝.分裂外推与组合技巧[M].北京:科学出版社,1998,35-58
[18]雷晋干,黄家鼎.算子方程投影解法[M].武汉大学出版社,1985,21-38
[19] R. Kress. A Nystro¨m method for boundary integral equations in domains with corners[J].Numer. Math.,1990,58:145-161
[20]朱瑞.三维轴对称问题的超收敛算法与分裂外推[D].成都:四川大学,2007,1-45
[21] M. A. Kelmanson. Solution of nonlinear elliptic equations with boundary singularities byan integral equation method[J]. J. Comput. Phys.,1984,56:244-283
[22] Y. S. Xu, Y. H. Zhao. An extrapolation method for a class of boundary integral equations[J].SIAM J. Math. Comp.,1996,65:587-610
[23] Y. Cao, X. He, T. Lu¨. A splitting extrapolation for solving nonlinear elliptic equations withd-quadratic finite elements[J]. SIAM J. Comp. Phy.,2009,1:109-122
[24] J. Helsing. Integral equation methods for elliptic problems with boundary conditions ofmixed type[J]. SIAM J. Comp. Phy.,2009,23:8892-8907
[25] A. S. Householder. The theory of matrices in numerical analysis[M]. Blaisdell PublishingCompany,1964,22-89
[26] C. Dagnino, V. Demichelis, P. Lamberti. A nodal spline collocation method for the solutionof cauchy singular integral equations[J]. J. Numer. Anal. Ind. App. Math.,2008,3:211-220
[27] P. M. Anselone. Singularity subtraction in the numerical solution of integral equations[J],J. Austral. Math. Soc.,1981,22:408-418
[28] C. A. Brebbia, S. Walker. Boundary elements techniques in engineering[M], Butter worthand Co., Boston,1980,1-30
[29] M. Willian. Strongly elliptic systems and boundary integral equations[M]. Cambridge Uni-versity Press,2000,22-68
[30] Y. Yan, I. Sloan. On integral equations of the first kind with logarithmic kernels[J]. J. Int.Eqns. Appl.,1988,1:517-548
[31] J. Saranen, I. Sloan. Quadrature methods for logarithmic-kernel integral equations on closedcurves[J]. IMA J. Numer. Anal.,1992,12:167-187
[32] J. Huang, Z. C. Li, I. L. Chen, et al. Advanced quadrature methods and splitting extrap-olation algorithms for first kind boundary integral equations of Laplace’s equation withdiscontinuity solutions[J]. Eng. Anal. Bound. Elem.,2010,34:1003-1008
[33] J. Huang, T. Lu¨, Z. C. Li. The mechanical quadrature methods and their splitting extrapo-lations for boundary integral equations of first kind on open arcs[J]. Appl. Numer. Math.,2009,59:2908-2922
[34] R. Zhu, J. Huang, T. Lu¨. Mechanical quadrature methods and their extrapolations forsolving boundary integral equations of axisymmetric Laplace mixed boundary value prob-lems[J]. Eng. Anal. Bound. Elem.,2006,30:391-398
[35] J. Saranen. The modified quadrature method for logarithmic-kernel integral equations onclosed curves[J]. J. Int. Eqns. Appl.,1991,3:575-600
[36] J. Elschner, I. Graham. An optimal order collocation method for first kind boundary integralequations on polygons[J]. Numer. Math.,1995,70:1-31
[37] G. Chandler. Galerkin’s method for boundary integral equations on polygonal domains[J].J. Aust. Math. Soc. Sor. B.V.,1984,26:1-13
[38] E. P. Stephan, W. L. Wendland. An augment Galerkin procedure for the boundary integralmethod applied to two-dimensional screen and crack problems[J]. Appl. Anal.,1984,18:183-219
[39] A. H. D. Cheng, D.T. Cheng. Heritage and early development of boundary elements. Eng.Anal. Bound. Elem.,2005,29:434-446
[40] C. B. Lin, T. Lu¨, T. M. Shih. The splitting extrapolation method. Singapore[M]. WorldScientific publiation,1995,1-60
[41] C. A. Brebbia, J. C. F. Telles, L. C. Wrobel. Boundary element techniques: Theory andapplications in engineering[M]. Berlin and New York, Springer-Verlag,1984,20-120
[42] C. A. Brebbia, L. C. Wrobel. The boundary element method[J]. Com. Meth. Fluids,1980,11:26-48
[43] C. A. Brebbia, D. Nardini. Dynamic analysis in solid mechanics by an alternative boundaryelement procedure[J]. Int. J. Soil Dyna. Earth. Eng.,1983,2(4):228-233
[44] C. A. Brebbia, P. Georgiou. Combination of boundary and finite elements in elastostatics[J].App. Math. Mode.,1979,3:212-220
[45] D. Nardini, C. A. Brebbia. A new approach to free vibration analysis using boundary ele-ments[J]. App. Math. Mode.,1983,7(3):157-162
[46] P. K. Banerjee. The boundary element methods in engineering[M]. London, McGraw-Hill,1994,1-60
[47] Z. C. Li. Combinations of method of fundamental solutions for Laplace’s equation withsingularities[J]. Eng. Anal. Bound. Elem.,2008,10:856-869
[48] S. Amini, S. P. Nixon. Preconditioned multiwavelet Galerkin boundary element solution ofLaplace’s equation[J]. Eng. Anal. Bound. Elem.,2006,7:523-530
[49] C. S. Liu. A modified collocation Trefftz method for the inverse Cauchy problem ofLaplace’s equation[J]. Eng. Anal. Bound. Elem.,2008,9:778-785
[50] A. Sidi, M. Israrli. Quadrature methods for periodic singular Fredholm integral equa-tions[J]. J. Sci. Comput.,1988,3:201-231
[51] J. Huang, Z. Wang. Extrapolation algorithm for solving mixed boundary integral equationsof the Helmholtz equation by mechanical quadrature methods[J]. SIAM J. Sci. Comput.,2009,6:4115-4129
[52] I. H. Sloan, A. Spence. The Galerkin method for integral equations of first-kind with loga-rithmic kernel: theory[J]. IMA J. Numer. Anal.,1988,8:105-122
[53] P. M. Anselone. Collectively compact operator approximation theory[M]. New Jersey,Prentice-Hall, Englewood Cliffs,1971,1-20
[54] F. Chatelin. Spectral approximation of linear operator[M]. Academic press,1983,20-50
[55] J. C. F. Telles, C. A. Brebbia. On the application of the boundary element method to plas-ticity Telles[J]. App. Math. Model.,1979,3:466-470
[56] J. C. F. Telles, C. A. Brebbia. Boundary element solution for half-plane problems[J]. Int.Jour. Sol. Str.,1981,17(12):1149-1158
[57] G. C. Hsiao，R. E. Maoacmy. Solution of bounded value Problems by integral equationsof the first kind[J]. SIAM Reriew,1973,15:687-705
[58] G. C. Hsiao, P. Kopp, W. L. Wendland. Some applications of a Galerkin-collocation methodfor boundary integral equations of the first kind[J]. Math. Meth. Appl. Sci.,1984,6(2):281-325
[59]江泽坚,孙善利.泛函分析[M].北京:高等教育出版社,2005,97-99
[60] G. C. Hsiao, J. Saranen. Boundary integral solution of the two-dimensional heat equation[J].Math. Meth. Appl. Sci.,1993,16(2):87–114
[61] G. C. Hsiao, W. gang, W. L. Wendland. A finite element method for some integral equationsof the first kind[J]. J. Math. Anal. App.,1977,58(3):449-481
[62]曾光.椭圆型方程的数值解法与稳定分析[D].成都:电子科技大学,2011,18-35
[63] W. L. Wendland, E. P. Stephan. A hypersingular boundary integral method for two-dimensional screen and crack problems[J]. Arch. Ratio. Mech. Anal.,1990,112(4):363-390
[64] W. L. Wendland, D. H. Yu. Adaptive boundary element methods for strongly elliptic integralequations[J]. Numer. Math.,1988,53(5):539-558
[65] G. Han, R. Wang. Richardson extrapolation of iterated discrete Galerkin solution for twodimensional Fredholm integral equations[J]. J. Comput. Appl. Math.,2002,139:49–63
[66] Q. Lin, I. H. Sloan, R. Xie. Extrapolation of the iterated-collocation method for integralequations of the second kind[J]. SIAM J. Numer. Anal.,1990,27:1535–1541
[67] T. Lin, Y. Lin, M. Rao, et al. Petrov–Galerkin methods for linear Volterra integral differ-ential equations[J]. SIAM J. Numer. Anal.,2000,38:937–963
[68] Y. Liu, T. Lu¨. Mechanical quadrature methods and their extrapolation for solving first kindAbel integral equations[J], J. Comput. Appl. Math.,2007,201:300–313
[69] F. G. Basombr′o, L. Guarracino, M. J. Ve′nere. A non-iterative algorithm based on Richard-son’s extrapolation: Application to groundwater flow modelling[J]. Int. J. Numer. MethodsEngrg.,2006,65:1088-1112
[70] B. Johansson, V. Christer. The use of Richardson extrapolation for the numerical solutionof low Mach number flow in confined regions[J]. J. Sci. Comput.,1993,8:307–340
[71] A. A. Bakr, R. T. Fenner. Boundary integral equation analysis of axisymmetric potentialproblems using isoparameter quadratic elements[R]. Technical report, Mech. Eng. Dept.,Imperical Collgeg,1981
[72] C. M. Chen, F. Liu, V. Anh. A Fourier method and an extrapolation technique forStokes’first problem for a heated generalized second grade fluid with fractional deriva-tive[J], J. Comput. Appl. Math.,2009,223:777–789
[73] M. Jung, U. Ru¨de. Implicit extrapolation methods for multilevel finite element computa-tions[J], SIAM J. Sci. Comput.,1996,17:156–179
[74] M. Jung, U. Ru¨de. Implicit extrapolation methods for variable coefficient problems[J],SIAM J. Sci. Comput.,1998,19:1109–1124
[75] M. D. Marcozzi. Extrapolation discontinuous Galerkin method for ultraparabolic equa-tions[J]. J. Comput. Appl. Math.,2009,224:679–687
[76] S. A. Richards. Completed Richardson extrapolation in space and time[J]. Comm. Numer.Methods Engrg.,1997,13:573–582
[77] A. Sidi. A complete convergence and stability theory for a generalized Richardson extrap-olation process[J]. SIAM J. Numer. Anal.,1997,34:1761–1778
[78] A. Sidi. The Richardson extrapolation process with a harmonic sequence of collocationpoints[J]. SIAM J. Numer. Anal.,2000,37:1729–1746
[79] Y. Wang, J. Zhang. Sixth order compact scheme combined with multigrid method and ex-trapolation technique for2D Poisson equation[J]. J. Comput. Phys.,2009,228:137–146
[80] G. Fairweather, Q. Lin, Y. Lin, J. Wang, S. Zhang. Asymptotic expansions and Richardsonextrapolation of approximate solutions for second order elliptic problems on rectangulardomains by mixed finite element methods[J]. SIAM J. Numer. Anal.,2006,44:1122–1149
[81] Q. Lin, J. Lin. Extrapolation of the bilinear element approximation for the Poisson equationon anisotropic meshes[J]. Numer. Meth. Par. Diff. Eqns.,2007,23:960–967
[82] Q. Lin, T. Lu¨, S. Shen. Maximum norm estimate extrapolation and optima points of stressesfor the finite element methods on the strongly triangulation[J]. J. Comput. Math.,1983,1:376–383
[83] J. Wang. Superconvergence and extrapolation for mixed finite element methods on rectan-gular domains[J], Math. Comp.,1991,56:477–503
[84] P. Neittaanma¨ki, Q. Lin. Acceleration of the convergence in finite difference methodsby predictor corrector and splitting extrapolation methods[J]. J. Comput. Math.,1987,5:181–190
[85] U. Ru¨de, A. Zhou. Multi-parameter extrapolation methods for boundary integral equa-tions. Numerical treatment of boundary integral equations[J]. Adv. Comput. Math.,1998,9:173–190
[86] U. Ru¨de. Book review: The Splitting Extrapolation Method[J]. SIAM Rev.,1997,39:161–162
[87] Y. Yan. The collocation method for first-kind boundary integral equations on polygonalregions[J]. Math. Comput.,1990,54:139-154
[88] G. I. Marchuk. Methods of Numerical Mathematies[M]. Springer-verlag, Berlin,1982,40-86
[89] R. Fo¨ssmeier. On Richardson extrapolation for finite difference methods on regular grids[J].Numer. Math.,1989,55:451–462
[90] G. I. Marchuk, V. V. ShaiDurov. Difference Methods and Their Extrapolations[J]. Appl.Math.,1983,22-65
[91] K. Rahul, S. N. Bhattacharyya. One-sided finite-difference approximations suitable for usewith Richardson extrapolation[J]. J. Comput. Phys.,2006,219:13–20
[92] H. Blum, Q. Lin, R. Rannacher. Asymptotic error expansion and Richardson extrapolationfor linear finite elements[J]. Numer. Math.,1986,49:11–37
[93] M. S. Abou El-seoud. Numerische behandlung von schwach singgularen integralgleichun-gen1. Art. Dissertation[M], Technische Hochschule Darmstadt, Federal Republic of Ger-many, Darmstadt,1979,1-53
[94] R. Bu¨rger, W. L. Wendland. Entropy boundary and jump conditions in the theory of sedi-mentation with compression[J]. Math. Meth. Appl. Sci.,1998,21(9):865–882
[95] J. L. Hess, A. M. O. Smith. Calculation of Potential Flow about Arbitrary Bodies[M], Aero-nautical Sciences, London,1967,120-169
[96] S. Christiansen. Numerical solution of an integral equation with a logarithmic kernel[J],BIT,1971,11(1):276-287
[97] J. C. Nedelec. Curved finite element methods for the solution of singular integral equationson surfaces in R3[J]. Comput. Meth. Appl. Mech. Eng.,1976,8(1):61-80
[98] J. Giroire, J. C. Nedelec. A new system of boundary integral equations for plates with freeedges[J]. Math. Meth. Appl. Sci.,1995,18(10):755–772
[99] H. Ammari, J. C. Nedelec. Small chirality behaviour of solutions to electromagnetic scat-tering problems in chiral media[J]. Math. Meth. Appl. Sci.,1998,21(4):327–359
[100] E. Becache, J.C. Nedelec, N. Nishimura. Regularization in3D for anisotropic elastody-namic crack and obstacle problems. Journal of Elasticity,1993,31(1):25-46.
[101] T. Lu¨，J. Huang. splitting extrapolation for solving the second order elliptic system withcurved boundary in Rdby using quadratic isoparametric finite element[J]. Appi. Numer.Math.,2002,40:467-481
[102] T. Lu¨，J. Huang. Quadrature methods with high accuracy and extrapolation for solvingboundary integral equations of first kind[J]. J. Math. Num. Sini.,2000, l:59-72
[103] J. Huang，T. Lu. SPlitting Extrapolations for solving Boundary integral equation of elas-ticity Diriehlet Problems on polygons[C]. Proeeeding of the third Intenational Confereceon boundary Element Techniques, Tsing hua University Press SPringe,2002:273-278
[104] G. A. Chandler. Super Convegence of Numerical Solutions to the seeond Kind IntegralEquation[D]. Australion National University，1979,55-98
[105] M. A. Jaswon，G. T. Symm. Intelgral Equation Methods in Potential Theory and Elasto-staties[M]. Aeademic Press,1977,192-208
[106] G. C. Hsiao，P. Kopp，W. L. Wendland. The synthesis of the collocation and Galerkinmethod applied to some integral equations of the first kind[J]. Devel. Bound. Elem. Meth.,1980,2:114-156
[107] A. A. Bakr. The boundary integral equation method in axisymmetric stress analysis prob-lems[M]. Springer-Verlag Berlin,1986,14:21-46
[108] I. H. Sloan，V. Thomee. Superconvergence of the Galerkin iterates for integral equationsof the seeond kind[J]. J. Integral Eqn.,1985, l-23
[109] J. Sarnen. Extrapolation methods for spline collocation solutions of pseudo-differentialequations on curves[J]. Numer. Math.,1989,56:385-407
[110] F. Russo Spena, A. Vacca. A Potential Formulation of Non-Linear Models of Flow throughAnisotropic Porous Media[J]. Trans. Porous Med.,2001,45:407-423
[111] M. Muskat. The Flow of Homogeneous Fluids through Porous Media[M]. Ann Arbor,Michigan,1972,345-365
[112] Y. Boubendir, S. Tlupova. Stokes-Darcy boundary integral solutions using precondition-ers[J]. J. Comput. Phys.,2009,228:8627-8641
[113] S. Tlupova, R. Cortez. Boundary integral solutions of coupled Stokes and Darcy flows[J].J. Comput. Phys.,2009,228:158-179
[114] L.-J. Qi, H.-S. Liao, P.-L. Su, S.-G. Li. The Technique Rotationa lControlVolume inNumerical Simulation of Anisotropic Seepage Flow[J]. J. Sichuan university,2008,1:10-15
[115] L. P. franca, C. Harder, F. Valentin. On a residual local projection method for the Darcyequation[J]. Comp. Ren. Math.,2009,17:1105-1110
[116] E. Ne′de′lec, J C. Approximation des e′quations integrals en me′lanique et on physique.Lecture notes, Centre de Mathe′matiques Applique′s[M], Ecole polytechnique, palaiseau,France,1977,76-92
[117] K. Atkinson. The numerical solution of Laplace’s equation on a wedge[J], IMA J Numer.Anal.,1984,4:19-41
[118] I. H. Sloan, A. Spence. The Galerkin method for integral equations of first-kind with loga-rithmic kernel: applications[J], IMA J. Numer. Anal.,1988,8:123-140
[119] B. Jacob. Dynamics of fluids in porous media[M]. American Elsevier Publishing Company,inc.,1972,365-390
[120] M. Georg, P. R. Jeffrey, L. W. Patricia, et al. Elements of Physical Hydrology[M]. JohnHopkins university Press,1998,12-58
[121] O. O. Okey. A Boundary Element-Finite Element Equation Solutions to Flow in Heteroge-neous Porous Media[J]. Trans. Por. Med.,1998,31:293-312
[122] C. Pozrikidis, D. A. Farrow. A model for fluid flow in solid tumors[J]. Ann. Biomed. Eng.,2003,31:181-194
[123] K. E. Atkinson, I. H. Sloan. The numerical solution of first-kind Logarithmic-kernel integralequation on smooth open arcs[J]. Math. Comp.,1991,56:119-139
[124] X. M. He, T. Lu¨. Splitting extrapolation method for solving second-order parabolic equa-tions with curved boundaries by using domain decomposition and d-quadratic isoparametricfinite elements[J]. Int. J. Comput. Math.,2007,84:767-781
[125] X. M. He, T. Lu¨. A finite element splitting extrapolation for second order hyperbolic equa-tions[J]. SIAM J. Sci. Comput.,2009,31:4244-4265
[126] I. Graham, Q. L., R. F. Xie. Extrapolation of Nystro¨m solutions of boundary integral equa-tions on non-smooth domains[J]. J. Comp. Math.,1992,10:231-244
[127] J. T. Chen, H. K. Hong, S. W. Chyuan. Boundary element analysis and design in seepageproblem using dual integral formulation[J]. Finite Elem. Anal. Des.,1994,17:1-20
[128] A. Mario, M. Gianmrco. Null space algorithm and spanning trees in solving Darcy’s equa-tion[J]. BIT Numer. Math.,2003,43:839–848
[129] P. Davis. Methods of Numerical Integration[M]. Academic Press, New York,1984,22-65
[130] A. Sidi. A new variable transformation for numerical integration[J]. Int. Ser. Numer. Math.,1983,112:359-373
[131] C. Z. Cheng, H. L. Zhou, Z. J. Hu, et al. Boundary element analysis of the seepage flow fieldat interior points close to the dam fundation[J]. J. University of Science and Technology ofChina,2006,12:1308-1313
[132] Y. Cao, X. M. He, T. Lu¨. An algorithm using the finite volume element method and itssplitting extrapolation[J]. J. Comput. Appl. Math.,2011,235:3734-3742
[133] Y. C. Shiah, C. L. Tan. BEM treatment of two-dimensional anisotropic field problems bydirect domain mapping[J]. Eng. Anal. Bound. Elem.,1997,20:347-351
[134] N. S. Mera, L. Elliott, D. B. Ingham, et al. A comparison of boundary element methodformulations for steady state anisotropic heat conduction problems[J]. Eng. Anal. Bound.Elem.,2001,25:115-128
[135] N. S. Mera, L. Elliott, D. B. Ingham, et al. An iterative bem for the cauchy steady stateheat conduction problem in an anisotropic medium with unknown thermal conductivitytensor[J]. Inverse Problems in Engineering,2000,8:579-607
[136] L. J. Segerlind. Applied finite element analysis[M]. John Wiley, New York,1984,150-187
[137] O. C. Zienkiewicz. The finite element method[M]. McGraw Hill, Maidenhead,1977,34-79
[138] Y. P. Chang, C. S. Kang, D. J. Chen. The use of fundamental Green’s functions for thesolution of heat conduction in anisotropic media[J]. Int. J. Heat Mass Trans.,1973,16:1905-1908
[139] F. Chatelin. Spectral approximation of linear operator[M]. Academic Press, New York,1983,118-186
[140] M. Costabel, V. J. Ervin, E. P. Stephan. On the convergence of collocation methods forSymm’s integral equation on open curves[J]. Math. Comp.,1988,51:167-179
[141] R. Kress. Linear integral equations[M]. Springer-Verlag, Belin,1989,62-117
[142] W. H. Li. Fluid mechanics in water resources engineering[M]. Allyn and Bacon, Toronto,1983,268-291
[143] J. J. Zhang, C. L. Tan, F. F. Afa. A general exact transformation of body-force volumeintegral in BEM for2D anisotropic elasticity[M]. Comput. Mech.,1996,19:1-10
[144] P. Davis. Circulant matrices[M]. New York, John Wiley and Sons,1979,88-125
[145] J. Huang, H. T. Huang, Z. C. Li, et al. Stability analysis via condition number and effectivecondition number for the first kind boundary integral equations by advanced quadraturemethods, a comparison. Engineering Analysis with Boundary Elements[J].2010,25:115-128
[146] K. Atkinson. The Numerical Solution of Integral Equations of the Second Kind[M]. Cam-bridge University Press, Cambridge,1997,25-63
[147]王元淳.边界元法基础[M].上海交通大学出版社,1988,1-50
[148] A. Sidi. Euler-Maclaurin expansions for integrals with end point sigularities:a new perpec-tive[J]. Numer. Math.,2004,98:371-387
[2]余德浩.自然边界方法的数学理论[M].北京:科学出版社,1993,1-18
[3]余德洁.无界区域上stokes问题的自然边界元与有限元素祸合法[J].计算数学，1992,14:371-378
[4]余德浩.无界区域非重叠型区域分解算法的离散化及其收敛性[J].计算数学,1996,3:328-336
[5]林群，朱超定.有限元的预处理和后处理理论[M].上海:科学技术出版社，1994,2-13
[6]朱起定，林群.有限元超收敛理论[M].湖南科技出版社，1989,18-28
[7]林群，严宁宁.高效有限元构造与分析[M].河北大学出版社,1996,35-65
[8]祝家麟.定常stokes问题的边界积分方程法[J].计算数学,1986,3:281–289
[9]祝家麟.椭圆边值问题的边界元分析[M].北京:科学出版社，1991,1-37
[10]祝家麟.边界元方法:一个老而新的发展中的数值方法[J].数学的实践与认识,1983,3:44-49
[11]吕涛，黄晋.平面双调和问题的第一类边界积分方程的高精度求积方法与外推[J].应用数学学报,2001,24(3):321-332
[12]吕涛，马长征.第二类边界积分方程nystro¨m解的高精度组合方法[J].计算物理,1994,11:75-84
[13]黄晋,吕涛.曲边多角形区域上第一类边界积分方程的高精度机械求积法与分裂外推[J].计算数学,2004,2:51–60
[14]黄晋,吕涛,申慧蓉.多角形域上Robin问题的边界积分方程的求积法与分裂外推[J].燕京大学学报,2004,2:137-140
[15]吕涛,黄晋.定常Stokes问题的边界积分方程的高精度求积方法与外推[J].计算物理,1999,16:561–567
[16]吕涛,黄晋.第二类若奇异积分方程的高精度Nystrom方法与外推[J].计算物理,1997,14(3):349–355
[17]吕涛,石济民,林振宝.分裂外推与组合技巧[M].北京:科学出版社,1998,35-58
[18]雷晋干,黄家鼎.算子方程投影解法[M].武汉大学出版社,1985,21-38
[19] R. Kress. A Nystro¨m method for boundary integral equations in domains with corners[J].Numer. Math.,1990,58:145-161
[20]朱瑞.三维轴对称问题的超收敛算法与分裂外推[D].成都:四川大学,2007,1-45
[21] M. A. Kelmanson. Solution of nonlinear elliptic equations with boundary singularities byan integral equation method[J]. J. Comput. Phys.,1984,56:244-283
[22] Y. S. Xu, Y. H. Zhao. An extrapolation method for a class of boundary integral equations[J].SIAM J. Math. Comp.,1996,65:587-610
[23] Y. Cao, X. He, T. Lu¨. A splitting extrapolation for solving nonlinear elliptic equations withd-quadratic finite elements[J]. SIAM J. Comp. Phy.,2009,1:109-122
[24] J. Helsing. Integral equation methods for elliptic problems with boundary conditions ofmixed type[J]. SIAM J. Comp. Phy.,2009,23:8892-8907
[25] A. S. Householder. The theory of matrices in numerical analysis[M]. Blaisdell PublishingCompany,1964,22-89
[26] C. Dagnino, V. Demichelis, P. Lamberti. A nodal spline collocation method for the solutionof cauchy singular integral equations[J]. J. Numer. Anal. Ind. App. Math.,2008,3:211-220
[27] P. M. Anselone. Singularity subtraction in the numerical solution of integral equations[J],J. Austral. Math. Soc.,1981,22:408-418
[28] C. A. Brebbia, S. Walker. Boundary elements techniques in engineering[M], Butter worthand Co., Boston,1980,1-30
[29] M. Willian. Strongly elliptic systems and boundary integral equations[M]. Cambridge Uni-versity Press,2000,22-68
[30] Y. Yan, I. Sloan. On integral equations of the first kind with logarithmic kernels[J]. J. Int.Eqns. Appl.,1988,1:517-548
[31] J. Saranen, I. Sloan. Quadrature methods for logarithmic-kernel integral equations on closedcurves[J]. IMA J. Numer. Anal.,1992,12:167-187
[32] J. Huang, Z. C. Li, I. L. Chen, et al. Advanced quadrature methods and splitting extrap-olation algorithms for first kind boundary integral equations of Laplace’s equation withdiscontinuity solutions[J]. Eng. Anal. Bound. Elem.,2010,34:1003-1008
[33] J. Huang, T. Lu¨, Z. C. Li. The mechanical quadrature methods and their splitting extrapo-lations for boundary integral equations of first kind on open arcs[J]. Appl. Numer. Math.,2009,59:2908-2922
[34] R. Zhu, J. Huang, T. Lu¨. Mechanical quadrature methods and their extrapolations forsolving boundary integral equations of axisymmetric Laplace mixed boundary value prob-lems[J]. Eng. Anal. Bound. Elem.,2006,30:391-398
[35] J. Saranen. The modified quadrature method for logarithmic-kernel integral equations onclosed curves[J]. J. Int. Eqns. Appl.,1991,3:575-600
[36] J. Elschner, I. Graham. An optimal order collocation method for first kind boundary integralequations on polygons[J]. Numer. Math.,1995,70:1-31
[37] G. Chandler. Galerkin’s method for boundary integral equations on polygonal domains[J].J. Aust. Math. Soc. Sor. B.V.,1984,26:1-13
[38] E. P. Stephan, W. L. Wendland. An augment Galerkin procedure for the boundary integralmethod applied to two-dimensional screen and crack problems[J]. Appl. Anal.,1984,18:183-219
[39] A. H. D. Cheng, D.T. Cheng. Heritage and early development of boundary elements. Eng.Anal. Bound. Elem.,2005,29:434-446
[40] C. B. Lin, T. Lu¨, T. M. Shih. The splitting extrapolation method. Singapore[M]. WorldScientific publiation,1995,1-60
[41] C. A. Brebbia, J. C. F. Telles, L. C. Wrobel. Boundary element techniques: Theory andapplications in engineering[M]. Berlin and New York, Springer-Verlag,1984,20-120
[42] C. A. Brebbia, L. C. Wrobel. The boundary element method[J]. Com. Meth. Fluids,1980,11:26-48
[43] C. A. Brebbia, D. Nardini. Dynamic analysis in solid mechanics by an alternative boundaryelement procedure[J]. Int. J. Soil Dyna. Earth. Eng.,1983,2(4):228-233
[44] C. A. Brebbia, P. Georgiou. Combination of boundary and finite elements in elastostatics[J].App. Math. Mode.,1979,3:212-220
[45] D. Nardini, C. A. Brebbia. A new approach to free vibration analysis using boundary ele-ments[J]. App. Math. Mode.,1983,7(3):157-162
[46] P. K. Banerjee. The boundary element methods in engineering[M]. London, McGraw-Hill,1994,1-60
[47] Z. C. Li. Combinations of method of fundamental solutions for Laplace’s equation withsingularities[J]. Eng. Anal. Bound. Elem.,2008,10:856-869
[48] S. Amini, S. P. Nixon. Preconditioned multiwavelet Galerkin boundary element solution ofLaplace’s equation[J]. Eng. Anal. Bound. Elem.,2006,7:523-530
[49] C. S. Liu. A modified collocation Trefftz method for the inverse Cauchy problem ofLaplace’s equation[J]. Eng. Anal. Bound. Elem.,2008,9:778-785
[50] A. Sidi, M. Israrli. Quadrature methods for periodic singular Fredholm integral equa-tions[J]. J. Sci. Comput.,1988,3:201-231
[51] J. Huang, Z. Wang. Extrapolation algorithm for solving mixed boundary integral equationsof the Helmholtz equation by mechanical quadrature methods[J]. SIAM J. Sci. Comput.,2009,6:4115-4129
[52] I. H. Sloan, A. Spence. The Galerkin method for integral equations of first-kind with loga-rithmic kernel: theory[J]. IMA J. Numer. Anal.,1988,8:105-122
[53] P. M. Anselone. Collectively compact operator approximation theory[M]. New Jersey,Prentice-Hall, Englewood Cliffs,1971,1-20
[54] F. Chatelin. Spectral approximation of linear operator[M]. Academic press,1983,20-50
[55] J. C. F. Telles, C. A. Brebbia. On the application of the boundary element method to plas-ticity Telles[J]. App. Math. Model.,1979,3:466-470
[56] J. C. F. Telles, C. A. Brebbia. Boundary element solution for half-plane problems[J]. Int.Jour. Sol. Str.,1981,17(12):1149-1158
[57] G. C. Hsiao，R. E. Maoacmy. Solution of bounded value Problems by integral equationsof the first kind[J]. SIAM Reriew,1973,15:687-705
[58] G. C. Hsiao, P. Kopp, W. L. Wendland. Some applications of a Galerkin-collocation methodfor boundary integral equations of the first kind[J]. Math. Meth. Appl. Sci.,1984,6(2):281-325
[59]江泽坚,孙善利.泛函分析[M].北京:高等教育出版社,2005,97-99
[60] G. C. Hsiao, J. Saranen. Boundary integral solution of the two-dimensional heat equation[J].Math. Meth. Appl. Sci.,1993,16(2):87–114
[61] G. C. Hsiao, W. gang, W. L. Wendland. A finite element method for some integral equationsof the first kind[J]. J. Math. Anal. App.,1977,58(3):449-481
[62]曾光.椭圆型方程的数值解法与稳定分析[D].成都:电子科技大学,2011,18-35
[63] W. L. Wendland, E. P. Stephan. A hypersingular boundary integral method for two-dimensional screen and crack problems[J]. Arch. Ratio. Mech. Anal.,1990,112(4):363-390
[64] W. L. Wendland, D. H. Yu. Adaptive boundary element methods for strongly elliptic integralequations[J]. Numer. Math.,1988,53(5):539-558
[65] G. Han, R. Wang. Richardson extrapolation of iterated discrete Galerkin solution for twodimensional Fredholm integral equations[J]. J. Comput. Appl. Math.,2002,139:49–63
[66] Q. Lin, I. H. Sloan, R. Xie. Extrapolation of the iterated-collocation method for integralequations of the second kind[J]. SIAM J. Numer. Anal.,1990,27:1535–1541
[67] T. Lin, Y. Lin, M. Rao, et al. Petrov–Galerkin methods for linear Volterra integral differ-ential equations[J]. SIAM J. Numer. Anal.,2000,38:937–963
[68] Y. Liu, T. Lu¨. Mechanical quadrature methods and their extrapolation for solving first kindAbel integral equations[J], J. Comput. Appl. Math.,2007,201:300–313
[69] F. G. Basombr′o, L. Guarracino, M. J. Ve′nere. A non-iterative algorithm based on Richard-son’s extrapolation: Application to groundwater flow modelling[J]. Int. J. Numer. MethodsEngrg.,2006,65:1088-1112
[70] B. Johansson, V. Christer. The use of Richardson extrapolation for the numerical solutionof low Mach number flow in confined regions[J]. J. Sci. Comput.,1993,8:307–340
[71] A. A. Bakr, R. T. Fenner. Boundary integral equation analysis of axisymmetric potentialproblems using isoparameter quadratic elements[R]. Technical report, Mech. Eng. Dept.,Imperical Collgeg,1981
[72] C. M. Chen, F. Liu, V. Anh. A Fourier method and an extrapolation technique forStokes’first problem for a heated generalized second grade fluid with fractional deriva-tive[J], J. Comput. Appl. Math.,2009,223:777–789
[73] M. Jung, U. Ru¨de. Implicit extrapolation methods for multilevel finite element computa-tions[J], SIAM J. Sci. Comput.,1996,17:156–179
[74] M. Jung, U. Ru¨de. Implicit extrapolation methods for variable coefficient problems[J],SIAM J. Sci. Comput.,1998,19:1109–1124
[75] M. D. Marcozzi. Extrapolation discontinuous Galerkin method for ultraparabolic equa-tions[J]. J. Comput. Appl. Math.,2009,224:679–687
[76] S. A. Richards. Completed Richardson extrapolation in space and time[J]. Comm. Numer.Methods Engrg.,1997,13:573–582
[77] A. Sidi. A complete convergence and stability theory for a generalized Richardson extrap-olation process[J]. SIAM J. Numer. Anal.,1997,34:1761–1778
[78] A. Sidi. The Richardson extrapolation process with a harmonic sequence of collocationpoints[J]. SIAM J. Numer. Anal.,2000,37:1729–1746
[79] Y. Wang, J. Zhang. Sixth order compact scheme combined with multigrid method and ex-trapolation technique for2D Poisson equation[J]. J. Comput. Phys.,2009,228:137–146
[80] G. Fairweather, Q. Lin, Y. Lin, J. Wang, S. Zhang. Asymptotic expansions and Richardsonextrapolation of approximate solutions for second order elliptic problems on rectangulardomains by mixed finite element methods[J]. SIAM J. Numer. Anal.,2006,44:1122–1149
[81] Q. Lin, J. Lin. Extrapolation of the bilinear element approximation for the Poisson equationon anisotropic meshes[J]. Numer. Meth. Par. Diff. Eqns.,2007,23:960–967
[82] Q. Lin, T. Lu¨, S. Shen. Maximum norm estimate extrapolation and optima points of stressesfor the finite element methods on the strongly triangulation[J]. J. Comput. Math.,1983,1:376–383
[83] J. Wang. Superconvergence and extrapolation for mixed finite element methods on rectan-gular domains[J], Math. Comp.,1991,56:477–503
[84] P. Neittaanma¨ki, Q. Lin. Acceleration of the convergence in finite difference methodsby predictor corrector and splitting extrapolation methods[J]. J. Comput. Math.,1987,5:181–190
[85] U. Ru¨de, A. Zhou. Multi-parameter extrapolation methods for boundary integral equa-tions. Numerical treatment of boundary integral equations[J]. Adv. Comput. Math.,1998,9:173–190
[86] U. Ru¨de. Book review: The Splitting Extrapolation Method[J]. SIAM Rev.,1997,39:161–162
[87] Y. Yan. The collocation method for first-kind boundary integral equations on polygonalregions[J]. Math. Comput.,1990,54:139-154
[88] G. I. Marchuk. Methods of Numerical Mathematies[M]. Springer-verlag, Berlin,1982,40-86
[89] R. Fo¨ssmeier. On Richardson extrapolation for finite difference methods on regular grids[J].Numer. Math.,1989,55:451–462
[90] G. I. Marchuk, V. V. ShaiDurov. Difference Methods and Their Extrapolations[J]. Appl.Math.,1983,22-65
[91] K. Rahul, S. N. Bhattacharyya. One-sided finite-difference approximations suitable for usewith Richardson extrapolation[J]. J. Comput. Phys.,2006,219:13–20
[92] H. Blum, Q. Lin, R. Rannacher. Asymptotic error expansion and Richardson extrapolationfor linear finite elements[J]. Numer. Math.,1986,49:11–37
[93] M. S. Abou El-seoud. Numerische behandlung von schwach singgularen integralgleichun-gen1. Art. Dissertation[M], Technische Hochschule Darmstadt, Federal Republic of Ger-many, Darmstadt,1979,1-53
[94] R. Bu¨rger, W. L. Wendland. Entropy boundary and jump conditions in the theory of sedi-mentation with compression[J]. Math. Meth. Appl. Sci.,1998,21(9):865–882
[95] J. L. Hess, A. M. O. Smith. Calculation of Potential Flow about Arbitrary Bodies[M], Aero-nautical Sciences, London,1967,120-169
[96] S. Christiansen. Numerical solution of an integral equation with a logarithmic kernel[J],BIT,1971,11(1):276-287
[97] J. C. Nedelec. Curved finite element methods for the solution of singular integral equationson surfaces in R3[J]. Comput. Meth. Appl. Mech. Eng.,1976,8(1):61-80
[98] J. Giroire, J. C. Nedelec. A new system of boundary integral equations for plates with freeedges[J]. Math. Meth. Appl. Sci.,1995,18(10):755–772
[99] H. Ammari, J. C. Nedelec. Small chirality behaviour of solutions to electromagnetic scat-tering problems in chiral media[J]. Math. Meth. Appl. Sci.,1998,21(4):327–359
[100] E. Becache, J.C. Nedelec, N. Nishimura. Regularization in3D for anisotropic elastody-namic crack and obstacle problems. Journal of Elasticity,1993,31(1):25-46.
[101] T. Lu¨，J. Huang. splitting extrapolation for solving the second order elliptic system withcurved boundary in Rdby using quadratic isoparametric finite element[J]. Appi. Numer.Math.,2002,40:467-481
[102] T. Lu¨，J. Huang. Quadrature methods with high accuracy and extrapolation for solvingboundary integral equations of first kind[J]. J. Math. Num. Sini.,2000, l:59-72
[103] J. Huang，T. Lu. SPlitting Extrapolations for solving Boundary integral equation of elas-ticity Diriehlet Problems on polygons[C]. Proeeeding of the third Intenational Confereceon boundary Element Techniques, Tsing hua University Press SPringe,2002:273-278
[104] G. A. Chandler. Super Convegence of Numerical Solutions to the seeond Kind IntegralEquation[D]. Australion National University，1979,55-98
[105] M. A. Jaswon，G. T. Symm. Intelgral Equation Methods in Potential Theory and Elasto-staties[M]. Aeademic Press,1977,192-208
[106] G. C. Hsiao，P. Kopp，W. L. Wendland. The synthesis of the collocation and Galerkinmethod applied to some integral equations of the first kind[J]. Devel. Bound. Elem. Meth.,1980,2:114-156
[107] A. A. Bakr. The boundary integral equation method in axisymmetric stress analysis prob-lems[M]. Springer-Verlag Berlin,1986,14:21-46
[108] I. H. Sloan，V. Thomee. Superconvergence of the Galerkin iterates for integral equationsof the seeond kind[J]. J. Integral Eqn.,1985, l-23
[109] J. Sarnen. Extrapolation methods for spline collocation solutions of pseudo-differentialequations on curves[J]. Numer. Math.,1989,56:385-407
[110] F. Russo Spena, A. Vacca. A Potential Formulation of Non-Linear Models of Flow throughAnisotropic Porous Media[J]. Trans. Porous Med.,2001,45:407-423
[111] M. Muskat. The Flow of Homogeneous Fluids through Porous Media[M]. Ann Arbor,Michigan,1972,345-365
[112] Y. Boubendir, S. Tlupova. Stokes-Darcy boundary integral solutions using precondition-ers[J]. J. Comput. Phys.,2009,228:8627-8641
[113] S. Tlupova, R. Cortez. Boundary integral solutions of coupled Stokes and Darcy flows[J].J. Comput. Phys.,2009,228:158-179
[114] L.-J. Qi, H.-S. Liao, P.-L. Su, S.-G. Li. The Technique Rotationa lControlVolume inNumerical Simulation of Anisotropic Seepage Flow[J]. J. Sichuan university,2008,1:10-15
[115] L. P. franca, C. Harder, F. Valentin. On a residual local projection method for the Darcyequation[J]. Comp. Ren. Math.,2009,17:1105-1110
[116] E. Ne′de′lec, J C. Approximation des e′quations integrals en me′lanique et on physique.Lecture notes, Centre de Mathe′matiques Applique′s[M], Ecole polytechnique, palaiseau,France,1977,76-92
[117] K. Atkinson. The numerical solution of Laplace’s equation on a wedge[J], IMA J Numer.Anal.,1984,4:19-41
[118] I. H. Sloan, A. Spence. The Galerkin method for integral equations of first-kind with loga-rithmic kernel: applications[J], IMA J. Numer. Anal.,1988,8:123-140
[119] B. Jacob. Dynamics of fluids in porous media[M]. American Elsevier Publishing Company,inc.,1972,365-390
[120] M. Georg, P. R. Jeffrey, L. W. Patricia, et al. Elements of Physical Hydrology[M]. JohnHopkins university Press,1998,12-58
[121] O. O. Okey. A Boundary Element-Finite Element Equation Solutions to Flow in Heteroge-neous Porous Media[J]. Trans. Por. Med.,1998,31:293-312
[122] C. Pozrikidis, D. A. Farrow. A model for fluid flow in solid tumors[J]. Ann. Biomed. Eng.,2003,31:181-194
[123] K. E. Atkinson, I. H. Sloan. The numerical solution of first-kind Logarithmic-kernel integralequation on smooth open arcs[J]. Math. Comp.,1991,56:119-139
[124] X. M. He, T. Lu¨. Splitting extrapolation method for solving second-order parabolic equa-tions with curved boundaries by using domain decomposition and d-quadratic isoparametricfinite elements[J]. Int. J. Comput. Math.,2007,84:767-781
[125] X. M. He, T. Lu¨. A finite element splitting extrapolation for second order hyperbolic equa-tions[J]. SIAM J. Sci. Comput.,2009,31:4244-4265
[126] I. Graham, Q. L., R. F. Xie. Extrapolation of Nystro¨m solutions of boundary integral equa-tions on non-smooth domains[J]. J. Comp. Math.,1992,10:231-244
[127] J. T. Chen, H. K. Hong, S. W. Chyuan. Boundary element analysis and design in seepageproblem using dual integral formulation[J]. Finite Elem. Anal. Des.,1994,17:1-20
[128] A. Mario, M. Gianmrco. Null space algorithm and spanning trees in solving Darcy’s equa-tion[J]. BIT Numer. Math.,2003,43:839–848
[129] P. Davis. Methods of Numerical Integration[M]. Academic Press, New York,1984,22-65
[130] A. Sidi. A new variable transformation for numerical integration[J]. Int. Ser. Numer. Math.,1983,112:359-373
[131] C. Z. Cheng, H. L. Zhou, Z. J. Hu, et al. Boundary element analysis of the seepage flow fieldat interior points close to the dam fundation[J]. J. University of Science and Technology ofChina,2006,12:1308-1313
[132] Y. Cao, X. M. He, T. Lu¨. An algorithm using the finite volume element method and itssplitting extrapolation[J]. J. Comput. Appl. Math.,2011,235:3734-3742
[133] Y. C. Shiah, C. L. Tan. BEM treatment of two-dimensional anisotropic field problems bydirect domain mapping[J]. Eng. Anal. Bound. Elem.,1997,20:347-351
[134] N. S. Mera, L. Elliott, D. B. Ingham, et al. A comparison of boundary element methodformulations for steady state anisotropic heat conduction problems[J]. Eng. Anal. Bound.Elem.,2001,25:115-128
[135] N. S. Mera, L. Elliott, D. B. Ingham, et al. An iterative bem for the cauchy steady stateheat conduction problem in an anisotropic medium with unknown thermal conductivitytensor[J]. Inverse Problems in Engineering,2000,8:579-607
[136] L. J. Segerlind. Applied finite element analysis[M]. John Wiley, New York,1984,150-187
[137] O. C. Zienkiewicz. The finite element method[M]. McGraw Hill, Maidenhead,1977,34-79
[138] Y. P. Chang, C. S. Kang, D. J. Chen. The use of fundamental Green’s functions for thesolution of heat conduction in anisotropic media[J]. Int. J. Heat Mass Trans.,1973,16:1905-1908
[139] F. Chatelin. Spectral approximation of linear operator[M]. Academic Press, New York,1983,118-186
[140] M. Costabel, V. J. Ervin, E. P. Stephan. On the convergence of collocation methods forSymm’s integral equation on open curves[J]. Math. Comp.,1988,51:167-179
[141] R. Kress. Linear integral equations[M]. Springer-Verlag, Belin,1989,62-117
[142] W. H. Li. Fluid mechanics in water resources engineering[M]. Allyn and Bacon, Toronto,1983,268-291
[143] J. J. Zhang, C. L. Tan, F. F. Afa. A general exact transformation of body-force volumeintegral in BEM for2D anisotropic elasticity[M]. Comput. Mech.,1996,19:1-10
[144] P. Davis. Circulant matrices[M]. New York, John Wiley and Sons,1979,88-125
[145] J. Huang, H. T. Huang, Z. C. Li, et al. Stability analysis via condition number and effectivecondition number for the first kind boundary integral equations by advanced quadraturemethods, a comparison. Engineering Analysis with Boundary Elements[J].2010,25:115-128
[146] K. Atkinson. The Numerical Solution of Integral Equations of the Second Kind[M]. Cam-bridge University Press, Cambridge,1997,25-63
[147]王元淳.边界元法基础[M].上海交通大学出版社,1988,1-50
[148] A. Sidi. Euler-Maclaurin expansions for integrals with end point sigularities:a new perpec-tive[J]. Numer. Math.,2004,98:371-387