基于ADI和IIM的界面问题算法研究
|
摘要
界面问题是自然界中一种常见的现象,对界面问题的数值方法研究在工业、生物、军事等方面有着重要的理论意义和实际应用价值,近些年一直受到学者们的广泛关注,也成为计算数学研究的前沿问题之一
首先,本文对界面问题数值方法的研究现状进行了综述.基于IIM方法,通过对传统的LOD差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维和三维热传导界面问题的LOD-IIM差分格式,所得差分格式在时间和空间方向以无穷范数均二阶收敛,并用三个数值实验验证了格式的收敛阶和稳定性.
其次,利用IIM方法的思想,通过对高阶紧致ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h3),而正则点处的局部截断误差仍为O(h4),构造了二维热传导界面问题的高阶紧致ADI差分格式,所得的差分格式以无穷范数在时间和空间方向分别为二阶和四阶收敛且无条件稳定.基于固定界面的热传导问题的解关于时间的连续性,采用Richardson外推方法将时间方向的精度提高到四阶,从而得到二维热传导界面问题在时间和空间方向均四阶收敛的差分格式.基于所构造的差分格式,对两个分片光滑的数值算例进行求解,验证了格式的收敛阶和无条件稳定性.
然后,采用Adams-Bashforth方法处理非线性对流项,通过对传统的ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维非线性对流扩散界面问题的ADI-IIM格式,所得格式在时间和空间方向以无穷范数均二阶收敛且无条件稳定.利用修正项关于时间的连续性,将当前时间层的修正项用前两个时间层上的修正项通过Adams-Bashforth方法逼近得到.对三个分片光滑的数值算例进行求解,验证了格式的收敛阶和稳定性以及通过Adams-Bashfoth方法逼近得到的修正项的可靠性.
最后,首次将Bernstein多项式用于椭圆界面问题数值方法的研究.在每个子区域上将方程的解表示为Bernstein多项式的线性组合,采用Galerkin方法或者配置方法计算Bernstein多项式的展开系数,得到一维椭圆界面问题和二维直线界面椭圆界面问题的高精度数值方法,数值实验表明该方法具有谱精度.对于二维曲线界面的椭圆界面问题,先通过一个非线性变换将其转化为直线界面问题并求解,再通过逆变换得到曲线界面问题的解.数值实验表明,该方法的逼近精度依赖于界面曲线的复杂程度.
Interface problems are ubiquitous in the nature. The research on numeri-cal methods for interface problems has great theoretical significance and actual application value in aspects of industry, biology and military. In recent years, authors have paid close attention to it and it has become one of the frontier prob-lems of computational mathematics.
Firstly, the thesis reviews the numerical methods for interfaces problems. Based on IIM, the LOD-IIM difference schemes for2D and3D heat interface problems are constructed by adding corrections at the right-hand side of differ-ence equations of traditional LOD schemes at irregular points so that the local truncation errors at irregular points are of O(h) and of O(h2) at regular points. Thus the LOD-IIM converges at the speed of two-order in both time and space in maximum norm. Three examples verify the convergence and unconditional stability.
Secondly, under the strategy of IIM, the HOC-ADI-IIM difference scheme is constructed for2D heat interface problems by adding corrections at the right-hand side of difference equations of traditional HOC-ADI scheme at irregular points so that the local truncation errors at irregular points are of O(h3) and of O(h4) at regular points. Thus the HOC-ADI-IIM scheme is unconditionally sta-ble and convergent two-order in time and four-order in space in maximum norm, respectively. Since the interface is fixed, the primary variables are continuous with time. The convergence order in time can be improved to be of four-order by Richardson extrapolation method. Two examples show that the scheme is unconditionally stable and convergent.
Then, by dealing the nonlinear convection term with Adams-Bashforth me-thod, the ADI-IIM scheme is constructed for2D nonlinear convection diffusion interface problems by adding corrections at the right-hand side of difference equations of traditional ADI scheme at irregular points so that the local trunca- tion errors at irregular points are of O(h) and of O(h2) at regular points. The proposed ADI-IIM scheme is two-order convergent in maximum norm in both time and space. Since the corrections are continuous with time, the corrections at present time level can be approximated by Adams-Bashfoth method using that from the two previous time levels. Three examples show that the proposed scheme is convergent and unconditionally stable and the approximated correc-tions are reliable.
Finally, the Bernstein polynomials are introduced to investigate numerical methods for elliptic interface problems for the first time. The solution on each subdomain is approximated by the linear combination of Bernstein polynomials. The expansion coefficients of Bernstein polynomials are computed by Galerkin methods or collocation methods. Thus the high accuracy numerical methods are obtained for1D elliptic interface problems and2D elliptic problems with rec-tilinear interfaces. Numerical examples shows that the proposed methods have spectral accuracy. For2D elliptic interface problems with curvilinear interfaces, one can firstly transform it into rectilinear interface problems by a nonlinear transformation and solve it. Then the numerical solutions for curvilinear inter-face problems can be obtained by the inverse transformation. Numerical exper-iments show that the accuracy of proposed method relies on the complexity of interface curve.
首先,本文对界面问题数值方法的研究现状进行了综述.基于IIM方法,通过对传统的LOD差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维和三维热传导界面问题的LOD-IIM差分格式,所得差分格式在时间和空间方向以无穷范数均二阶收敛,并用三个数值实验验证了格式的收敛阶和稳定性.
其次,利用IIM方法的思想,通过对高阶紧致ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h3),而正则点处的局部截断误差仍为O(h4),构造了二维热传导界面问题的高阶紧致ADI差分格式,所得的差分格式以无穷范数在时间和空间方向分别为二阶和四阶收敛且无条件稳定.基于固定界面的热传导问题的解关于时间的连续性,采用Richardson外推方法将时间方向的精度提高到四阶,从而得到二维热传导界面问题在时间和空间方向均四阶收敛的差分格式.基于所构造的差分格式,对两个分片光滑的数值算例进行求解,验证了格式的收敛阶和无条件稳定性.
然后,采用Adams-Bashforth方法处理非线性对流项,通过对传统的ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维非线性对流扩散界面问题的ADI-IIM格式,所得格式在时间和空间方向以无穷范数均二阶收敛且无条件稳定.利用修正项关于时间的连续性,将当前时间层的修正项用前两个时间层上的修正项通过Adams-Bashforth方法逼近得到.对三个分片光滑的数值算例进行求解,验证了格式的收敛阶和稳定性以及通过Adams-Bashfoth方法逼近得到的修正项的可靠性.
最后,首次将Bernstein多项式用于椭圆界面问题数值方法的研究.在每个子区域上将方程的解表示为Bernstein多项式的线性组合,采用Galerkin方法或者配置方法计算Bernstein多项式的展开系数,得到一维椭圆界面问题和二维直线界面椭圆界面问题的高精度数值方法,数值实验表明该方法具有谱精度.对于二维曲线界面的椭圆界面问题,先通过一个非线性变换将其转化为直线界面问题并求解,再通过逆变换得到曲线界面问题的解.数值实验表明,该方法的逼近精度依赖于界面曲线的复杂程度.
Interface problems are ubiquitous in the nature. The research on numeri-cal methods for interface problems has great theoretical significance and actual application value in aspects of industry, biology and military. In recent years, authors have paid close attention to it and it has become one of the frontier prob-lems of computational mathematics.
Firstly, the thesis reviews the numerical methods for interfaces problems. Based on IIM, the LOD-IIM difference schemes for2D and3D heat interface problems are constructed by adding corrections at the right-hand side of differ-ence equations of traditional LOD schemes at irregular points so that the local truncation errors at irregular points are of O(h) and of O(h2) at regular points. Thus the LOD-IIM converges at the speed of two-order in both time and space in maximum norm. Three examples verify the convergence and unconditional stability.
Secondly, under the strategy of IIM, the HOC-ADI-IIM difference scheme is constructed for2D heat interface problems by adding corrections at the right-hand side of difference equations of traditional HOC-ADI scheme at irregular points so that the local truncation errors at irregular points are of O(h3) and of O(h4) at regular points. Thus the HOC-ADI-IIM scheme is unconditionally sta-ble and convergent two-order in time and four-order in space in maximum norm, respectively. Since the interface is fixed, the primary variables are continuous with time. The convergence order in time can be improved to be of four-order by Richardson extrapolation method. Two examples show that the scheme is unconditionally stable and convergent.
Then, by dealing the nonlinear convection term with Adams-Bashforth me-thod, the ADI-IIM scheme is constructed for2D nonlinear convection diffusion interface problems by adding corrections at the right-hand side of difference equations of traditional ADI scheme at irregular points so that the local trunca- tion errors at irregular points are of O(h) and of O(h2) at regular points. The proposed ADI-IIM scheme is two-order convergent in maximum norm in both time and space. Since the corrections are continuous with time, the corrections at present time level can be approximated by Adams-Bashfoth method using that from the two previous time levels. Three examples show that the proposed scheme is convergent and unconditionally stable and the approximated correc-tions are reliable.
Finally, the Bernstein polynomials are introduced to investigate numerical methods for elliptic interface problems for the first time. The solution on each subdomain is approximated by the linear combination of Bernstein polynomials. The expansion coefficients of Bernstein polynomials are computed by Galerkin methods or collocation methods. Thus the high accuracy numerical methods are obtained for1D elliptic interface problems and2D elliptic problems with rec-tilinear interfaces. Numerical examples shows that the proposed methods have spectral accuracy. For2D elliptic interface problems with curvilinear interfaces, one can firstly transform it into rectilinear interface problems by a nonlinear transformation and solve it. Then the numerical solutions for curvilinear inter-face problems can be obtained by the inverse transformation. Numerical exper-iments show that the accuracy of proposed method relies on the complexity of interface curve.
引文
[1]Z. Li and M. C. Lai. The immersed interface method for the Navier-Stokes equations with singular forces [J]. J. Comput. Phys.,2001,171(2):822-842.
[2]Z. Li. An overview of the immersed interface method and its applications [J]. Taiwanese J. Math.,2003,7(1):1-49.
[3]A. McKenney, L. Greengard, and A. Mayo. A fast poisson solver for complex geometries [J]. J. Comput. Phys.,1995,118(2):348-355.
[4]R. J. LeVeque. Clawpack and amrclaw-software for high-resolution godunov methods [C]. Golden Colorado.
[5]S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations [J].J. Comput. Phys.,1988,79(l):12-49.
[6]L. Adams. A multigrid algorithm for immersed interface problems [C]. Citeseer,1996, pages 1-14.
[7]P. M. De Zeeuw. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver [J]. J. Comput. Appl. Math.,1990,33(1):1-27.
[8]C. S. Peskin. The immersed boundary method [J]. Acta Numer.,2002,11(1):479-517.
[9]M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to in-compressible two-phase flow [J]. J. Comput. Phys.,1994,114:146-159.
[10]Z. Li and K. Ito. The immersed interface method:numerical solutions of PDEs involving interfaces and irregular domains [M]. Soc. Ind. Appl. Math.,2006.
[11]M. Kumar and P. Joshi. Some numerical techniques for solving elliptic interface problems [J]. Numerical Methods for Partial Differential Equations,2010.
[12]J. B. Bell, C. N. Dawson, and G. R. Shubin. An unsplit, higher order godunov method for scalar conservation laws in multiple dimensions [J]. J. Comput. Phys.,1988,74(1):1-24.
[13]G. R. Shubin and J. B. Bell. An analysis of the grid orientation effect in numerical simulation of miscible displacement [J]. Comput. Methods Appl. Mech. Engrg.,1984,47(1-2):47-71.
[14]Z. Li. The Immersed Interface Method:A Numerical Approach for Partial Differential Equa-tions with Interfaces [D]. Seattle Washington, University of Washington,1994.
[15]C. S. Peskin. Flow patterns around heart valves:a numerical method [J]. J. Comput. Phys., 1972,10(2):252-271.
[16]C. S. Peskin. Numerical analysis of blood flow in the heart [J].J. Comput. Phys.,1977, 25(3):220-252.
[17]C. S. Peskin and D. M. McQueen. A three-dimensional computational method for blood flow in the heart. I. Immersed elastic fibers in a viscous incompressible fluid [J]. J. Comput. Phys., 1989,81(2):372-405.
[18]D. M. McQueen and C. S. Peskin. A three-dimensional computational method for blood flow in the heart. II. Contractile fibers [J]. J. Comput. Phys.,1989,82(2):289-297.
[19]R. P. Beyer Jr. A computational model of the cochlea using the immersed boundary method [J]. J. Comput. Phys.,1992,98(1):145-162.
[20]E. Givelberg. Modeling elastic shells immersed in fluid [J]. Commun. Pure Appl. Math.,2004, 57(3):283-309.
[21]A. L. Fogelson. A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting [J]. J. Comput. Phys.,1984,56(1):111-134.
[22]N. T. Wang and A. L. Fogelson. Computational Methods for Continuum Models of Platelet Aggregation [J]. J. Comput. Phys.,1999,151(2):649-675.
[23]L. J. Fauci and C. S. Peskin. A computational model of aquatic animal locomotion [J]. J. Comput. Phys.,1988,77(l):85-108.
[24]M. M. Hopkins and L. J. Fauci. A computational model of the collective fluid dynamics of motile micro-organisms [J]. J. Fluid Mech.,2002,455:149-174.
[25]R. Cortez, L. Fauci, N. Cowen, and R. Dillon. Simulation of swimming organisms:Coupling internal mechanics with external fluid dynamics [J]. Comput. Sci. Eng., IEEE,2005,6(3):38-45.
[26]L. Zhu and C. S. Peskin. Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method [J]. J. Comput. Phys.,2002,179(2):452-468.
[27]J. M. Stockie and S. I. Green. Simulating the Motion of Flexible Pulp Fibres Using the Im-mersed Boundary Method [J]. J. Comput. Phys.,1998,147(1):147-165.
[28]M. Uhlmann. An immersed boundary method with direct forcing for the simulation of partic-ulate flows [J]. J. Comput. Phys.,2005,209(2):448-476.
[29]L. A. Miller and C. S. Peskin. When vortices stick:an aerodynamic transition in tiny insect flight [J]. J. Exp. Biol.,2004,207(17):3073-3088.
[30]L. A. Miller and C. S. Peskin. A computational fluid dynamics of 'clap and fling' in the smallest insects [J]. J. Exp. Biol.,2005,208(2):195-212.
[31]R. J. Leveque and Z. Li. The immersed interface method for elliptic equations with discontin-uous coefficients and singular sources [J]. SIAM J. Numer. Anal.,1994,31(4):1019-1044.
[32]Z. Li and K. Ito. Maximum Principle Preserving Schemes for Interface Problems [J]. SIAM J. Sci. Comput.,2001,23:339-361.
[33]L. Adams and Z. Li. The immersed interface/multigrid methods for interface problems [J]. SIAM J. Sci. Comput.,2003,24(2):463-479.
[34]Z. Li. A fast iterative algorithm for elliptic interface problems [J]. SIAM J. Numer. Anal.,1998, 35(1):230-254.
[35]A. Mayo. The fast solution of Poisson's and the biharmonic equations on irregular regions [J]. SIAM J. Numer. Anal.,1984,21(2):285-299.
[36]A. Wiegmann and K. P. Bube. The explicit-jump immersed interface method:finite differ-ence methods for PDEs with piecewise smooth solutions [J]. SIAM J. Numer. Anal.,2000, 37(3):827-862.
[37]P. A. Berthelsen. A decomposed immersed interface method for variable coefficient ellip-tic equations with non-smooth and discontinuous solutions [J]. J. Comput. Phys.,2004, 197(1):364-386.
[38]P. Joshi and M. Kumar. Mathematical model and computer simulation of three dimensional thin film elliptic interface problems [J]. Comput. Math. Appl,2011,63(1):25-35.
[39]H. Huang and Z. Li. Convergence analysis of the immersed interface method [J]. IMA J. Numer. Anal.,1999,19(4):583-608.
[40]J. T. Beale and A. T. Layton. On the accuracy of finite difference methods for elliptic problems with interfaces [J]. Commun. Appl. Math. Comput. Sci.,2006,1(1):91-119.
[41]B.A. Vanderlei, M.M. Hopkins, and L.J. Fauci. Error estimation for immersed interface solu-tions [C].2007, pages 115-125.
[42]C. Zhang and R. J. LeVeque. The immersed interface method for acoustic wave equations with discontinuous coefficients [J]. Wave motion,1997,25(3):237-263.
[43]C. Zhang. Immersed interface methods for hyperbolic systems of partial differential equations with discontinuous coefficients [D]. University of Washington,1996.
[44]J. Piraux and B. Lombard. A new interface method for hyperbolic problems with discontinuous coefficients:one-dimensional acoustic example [J]. J. Comput. Phys.,2001,168(1):227-248.
[45]I. Brayanov, J. Kandilarov, and M. Koleva. Immersed interface difference schemes for a parabolic-elliptic interface problem [J]. Large-Scale Sci. Comput.,2008, pages 661-669.
[46]Z. Li and A. Mayo. ADI methods for heat equations with discontinuities along an arbitrary interface [C]. American Mathematical Society,1994,48:311-315.
[47]Z. Li and B. Soni. Fast and accurate numerical approaches for Stefan problems and crystal growth [J]. Numer. Heat TR. B-Fund.,1999,35(4):461-484.
[48]Z. Li. Immersed interface methods for moving interface problems [J]. Numer. Algorithms, 1997,14(4):269-293.
[49]A. Wiegmann and K. P. Bube. The immersed interface method for nonlinear differential equa-tions with discontinuous coefficients and singular sources [J]. SIAM J. Numer. Anal.,1998, 35(1):177-200.
[50]J. D. Kandilarov and L. G. Vulkov. The immersed interface method for two-dimensional heat-diffusion equations with singular own sources [J]. Appl. Numer. Math.,2007,57(5-7):486-497.
[51]Z. Li, D. Wang, and J. Zou. Theoretical and numerical analysis on a thermo-elastic system with discontinuities [J]. J. Comput. Appl. Math.,1998,92(1):37-58.
[52]I. Babuska. The finite element method for elliptic equations with discontinuous coefficients [J]. Computing,1970,5(3):207-213.
[53]J. H. Bramble and J. T. King. A finite element method for interface problems in domains with smooth boundaries and interfaces [J]. Adv. Comput. Math.,1996,6(1):109-138.
[54]J.W. Barrett and C.M. Elliott. Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces [J]. IMA J. Numer. Anal.,1987,7(3):283-300.
[55]C. Bernardi and R. Verfiirth. Adaptive finite element methods for elliptic equations with non-smooth coefficients [J]. Numer. Math.,2000,85(4):579-608.
[56]Z. Chen and S. Dai. On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients [J]. SIAM J. Sci. Comput.,2002,24(2):443-462.
[57]J. W. Barrett and C. M. Elliott. Finite element approximation of the Dirichlet problem using the boundary penalty method [J]. Numer. Math.,1986,49(4):343-366.
[58]R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids [J]. ESAIM-Math. Model. Numer. Anal.,2003,37(2):209-226.
[59]J. H. Bramble and J. T. King. A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries [J]. Math. Comp.,1994,63(207):1-18.
[60]Z. Chen and J. Zou. Finite element methods and their convergence for elliptic and parabolic interface problems [J]. Numer. Math.,1998,79(2):175-202.
[61]M. Feistauer and V. Sobotikova. Finite element approximation of nonlinear elliptic problems with discontinuous coefficients [J]. Comment. Math. Univ. Carolin.,1989,30(1):457-500.
[62]SP Marin. A finite element method for exterior interface problems [J]. Int. J. Math. Math. Sci., 1980,3:311-350.
[63]Z. Li. The immersed interface method using a finite element formulation [J]. Appl. Numer. Math.,1998,27(3):253-267.
[64]Z. Li, T. Lin, and X. Wu. New cartesian grid methods for interface problems using the finite element formulation [J]. Numer. Math.,2003,96(1):61-98.
[65]Y. Gong, B. Li, and Z. Li. Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions [J]. SIAM J. Numer. Anal,2008,46(1):472-495.
[66]Z. Li, T. Lin, Y. Lin, and RC Rogers. An immersed finite element space and its approximation capability [J]. Numer. Methods Partial Differential Equations,2004,20(3):338-367.
[67]L. Zhang, A. Gerstenberger, X. Wang, and W.K. Liu. Immersed finite element method [J]. Comput. Methods Appl. Mech. Engrg.,2004,193(21):2051-2067.
[68]X. He, T. Lin, and Y. Lin. Approximation capability of a bilinear immersed finite element space [J]. Numer. Methods Partial Differential Equations,2008,24(5):1265-1300.
[69]B. Camp, T. Lin, Y. Lin, and W. Sun. Quadratic immersed finite element spaces and their approximation capabilities [J]. Adv. Comput. Math.,2006,24(1):81-112.
[70]X. He, T. Lin, and Y. Lin. The convergence of the bilinear and linear immersed finite ele-ment solutions to interface problems [J]. Numer. Methods Partial Differential Equations,2012, 28(1):312-330.
[71]A. Hansbo and P. Hansbo. An unfitted finite element method, based on nitsche's method, for elliptic interface problems [J]. Comput. Methods Appl. Mech. Engrg.,2002,191(47):5537-5552.
[72]A. Loubenets, T. Ali, and M. Hanke. Highly accurate finite element method for one-dimensional elliptic interface problems [J]. Appl. Numer. Math.,2009,59(1):119-134.
[73]S. Adjerid and T. Lin. A p-th degree immersed finite element for boundary value problems with discontinuous coefficients [J]. Appl. Numer. Math.,2009,59(6):1303-1321.
[74]T. Lin, Y. Lin, W. W. Sun, and Z. Wang. Immersed finite element methods for 4th order differential equations [J]. J. Comput. Appl. Math.,2011,235(13):3953-3964.
[75]L. Mu, J. Wang, G. Wei, X. Ye, and S. Zhao. Weak galerkin methods for second order elliptic interface problems [J]. Arxiv preprint arXiv:1201.6438,2012.
[76]W. K. Liu, Y. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui, N. Patankar, Y. Zhang, C. Bajaj, J. Lee, et al. Immersed finite element method and its applications to biological systems [J]. Comput. Methods Appl. Mech. Engrg.,2006,195(13):1722-1749.
[77]J. Huang and J. Zou. Some new a priori estimates for second-order elliptic and parabolic interface problems [J]. J. Differential Equations,2002,184(2):570-586.
[78]W. K. Liu, D. W. Kim, and S. Tang. Mathematical foundations of the immersed finite element method [J]. Comput. Mech.,2007,39(3):211-222.
[79]L. T. Zhang and M. Gay. Immersed finite element method for fluid-structure interactions [J]. J. Fluid. Struct.,2007,23(6):839-857.
[80]S. H. Chou, D. Y. Kwak, and K. T. Wee. Optimal convergence analysis of an immersed interface finite element method [J]. Adv. Comput. Math.,2010,33(2):149-168.
[81]S. Wang and H. Chen. The optimal convergence analysis for an immersed finite element method [C]. IEEE,2011, pages 255-258.
[82]S. Adjerid and T. Lin. Higher-order immersed discontinuous galerkin methods [J]. Int. J. Inf. Syst. Sci.,2007,3(4):555-568.
[83]C. Engwer, P. Bastian, and S. P. Kuttanikkad. An unfitted discontinuous galerkin finite element method for pore scale simulations [C].2008,6126.
[84]P. Bastian, C. Engwer, J. Fahlke, and O. Ippisch. An unfitted discontinuous galerkin method for pore-scale simulations of solute transport [J]. Math. Comput. Simulation,2011, 81(10):2051-2061.
[85]X. He, T. Lin, and Y. Lin. Interior penalty bilinear ife discontinuous galerkin methods for elliptic equations with discontinuous coefficient [J]. J. Syst. Sci. Complex.,2010,23(3):467-483.
[86]F. Heimann, C. Engwer, O. Ippisch, and P. Bastian. An unfitted interior penalty discontinu-ous galerkin method for incompressible navier-stokes two-phase flow [J]. Internal J. Numer. Methods Fluids,2012.
[87]H. Wu and Y. Xiao. An unfitted hp-interface penalty finite element method for elliptic interface problems [J]. Arxiv preprint arXiv:1007.2893,2010.
[88]Y. R. Efendiev and X. H. Wu. Multiscale finite element for problems with highly oscillatory coefficients [J]. Numer. Math.,2002,90(3):459-486.
[89]CC Chu, I.G. Graham, and TY Hou. A new multiscale finite element method for high-contrast elliptic interface problems [J]. Math. Comp.,2010,79(272):1915-1955.
[90]R. R. Millward. A new adaptive multiscale finite element method with applications to high contrast interface problems [D]. University of Bath,2011.
[91]A. Liang, X. Jing, and X. Sun. Constructing spectral schemes of the immersed interface method via a global description of discontinuous functions [J]. J. Comput. Phys.,2008,227(18):8341-8366.
[92]Y. JIANG, A. LIANG, X. SUN, and X. JING. Multi-domain spectral immersed interface method for solving elliptic equation with a global description of discontinuous functions [J]. Chinese J. Aeronaut.,2012,25(3):297-310.
[93]M. N. Linnick and H. F. Fasel. A high-order immersed interface method for simulating un-steady incompressible flows on irregular domains [J].J. Comput. Phys.,2005,204(1):157-192.
[94]X. Zhong. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity [J]. J. Comput. Phys.,2007,225(1):1066-1099.
[95]I.T. Angelova and L.G. Vulkov. High-order finite difference schemes for elliptic problems with intersecting interfaces [J]. Appl. Math. Comput.,2007,187(2):824-843.
[96]X. D. Liu, R. P. Fedkiw, and M. Kang. A boundary condition capturing method for Poisson's equation on irregular domains [J]. J. Comput. Phys.,2000,160(l):151-178.
[97]X.D. Liu and T.C. Sideris. Convergence of the ghost fluid method for elliptic equations with interfaces [J]. Math. Comp.,2003,72(244):1731-1746.
[98]M. Kang, R. P. Fedkiw, and X. D. Liu. A boundary condition capturing method for multiphase incompressible flow [J]. J. Sci. Comput.,2000,15(3):323-360.
[99]D. Q. Nguyen, R. P. Fedkiw, and M. Kang. A boundary condition capturing method for incom-pressible flame discontinuities [J].J. Comput. Phys.,2001,172(1):71-98.
[100]D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle level set method for improved interface capturing [J]. J. Comput. Phys.,2002,183(1):83-116.
[101]F. Gibou, R. P. Fedkiw, L. T. Cheng, and M. Kang. A second-order-accurate symmetric discretization of the poisson equation on irregular domains [J]. J. Comput. Phys.,2002, 176(1):205-227.
[102]B. C. Shin and J. H. Jung. Spectral collocation and radial basis function methods for one-dimensional interface problems [J]. Appl. Numer. Math.,2011,61 (8):911-928.
[103]A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome. A conservative adaptive projection method for the variable density incompressible navier-stokes equations [J]. J. Comput. Phys.,1998,142(1):1-46.
[104]A. S. Almgren, J. B. Bell, P. Colella, and T. Marthaler. A cartesian grid projection method for the incompressible Euler equations in complex geometries [J]. SIAM J. Sci. Comput.,1997, 18(5):1289-1309.
[105]R. Ewing, O. Iliev, and R. Lazarov. A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients [J]. SIAM J. Sci. Comput.,2001,23(4):1335-1351.
[106]H. Johansen and P. Colella. A cartesian grid embedded boundary method for Poisson's equation on irregular domains [J]. J. Comput. Phys.,1998,147(1):60-85.
[107]R. E. Ewing, Z. Li, T. Lin, and Y. Lin. The immersed finite volume element methods for the elliptic interface problems [J]. Math. Comput. Simulation,1999,50(l):63-76.
[108]A. Naumovich. On a finite volume discretization of the three-dimensional biot poroelasticity system in multilayered domains [J]. Comput. Methods Appl. Math.,2006,6(3):306.
[109]XM He, T. Lin, and Y. Lin. A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficient [J]. Commun. Comput. Phys.,2010,6(1):185-202.
[110]M. Oevermann, C. Scharfenberg, and R. Klein. A sharp interface finite volume method for elliptic equations on cartesian grids [J]. J. Comput. Phys.,2009,228(14):5184-5206.
[111]A. Mayo. A decomposition finite difference method for the fourth order accurate solution of Poisson's equation on general regions [J]. Int. J. High Speed Comput.,1991,3:89-106.
[112]A. Mayo. The rapid evaluation of volume integrals of potential theory on general regions [J]. J. Comput. Phys.,1992,100(2):236-245.
[113]A. Greenbaum, L. Greengard, and A. Mayo. On the numerical solution of the biharmonic equation in the plane [J]. Phys. D,1992,60(1-4):216-225.
[114]A. Mayo. High order accurate particular solutions of the biharmonic equation on general regions [J]. Contemp. Math.,2003,323:233-244.
[115]A. Mayo and A. Greenbaum. Fast parallel iterative solution of poisson's and the biharmonic equations on irregular regions [J]. SIAM J. Sci. Statist. Comput.,1992,13(1):101-118.
[116]L. Greengard, M.C. Kropinski, and A. Mayo. Integral equation methods for Stokes flow and isotropic elasticity in the plane [J]. J. Comput. Phys.,1996,125(2):403-414.
[117]A. Mayo. Rapid, fourth order accurate solution of the steady Navier Stokes equations on general regions [J]. Dyn. Contin. Discrete Impuls. Syst.,2005,12:59-72.
[118]YC Zhou, S. Zhao, M. Feig, and GW Wei. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources [J]. J. Comput. Phys., 2006,213(1):1-30.
[119]SN Yu, Y. Xiang, and GW Wei. Matched interface and boundary (MIB) method for the vibra-tion analysis of plates [J]. Comm. Numer. Methods Engrg.,2009,25(9):923-950.
[120]S. Yu, Y. Zhou, and G. W. Wei. Matched interface and boundary (mib) method for elliptic problems with sharp-edged interfaces [J].J. Comput. Phys.,2007,224(2):729-756.
[121]S. Yu and G. W. Wei. Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities [J]. J. Comput. Phys.,2007,227(1):602-632.
[122]YC Zhou, J. Liu, and D.L. Harry. A matched interface and boundary method for solving multi-flow navier-stokes equations with applications to geodynamics [J]. Journal of Computational Physics,2011,231(1):223-242.
[123]S. Zhao and G. W. Wei. Matched interface and boundary (mib) for the implementation of boundary conditions in high-order central finite differences [J]. Internat. J. Numer. Methods Engrg.,2009,77(12):1690-1730.
[124]S. Zhao. High order matched interface and boundary methods for the helmholtz equation in media with arbitrarily curved interfaces [J]. J. Comput. Phys.,2010,229(9):3155-3170.
[125]K. Pan, Y. Tan, and H. Hu. An interpolation matched interface and boundary method for elliptic interface problems [J]. J. Comput. Appl. Math.,2010,234(l):73-94.
[126]I. Chern and Y. Shu. A coupling interface method for elliptic interface problems [J]. J. Comput. Phys.,2007,225(2):2138-2174.
[127]Y. C. Shu, C. Y. Kao, I. Chern, and C. C. Chang. Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients [J]. J. Comput. Phys.,2010, 229(24):9246-9268.
[128]J. Kandilarov. A coupling interface method for a nonlinear parabolic-elliptic problem [J]. Numer. Anal. Appl.,2009, pages 330-337.
[129]H. Han, Z. Huang, and R.B. Kellogg. A tailored finite point method for a singular perturbation problem on an unbounded domain [J]. J. Sci. Comput.,2008,36(2):243-261.
[130]H. Han and Z. Huang. Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions [J]. J. Sci. Comput.,2009,41(2):200-220.
[131]Z. Huang. Tailored finite point method for the interface problem [J]. Netw. Heterog. Media, 2009,4(1):91-106.
[132]Y. Shih, R.B. Kellogg, and P. Tsai. A tailored finite point method for convection-diffusion-reaction problems [J]. J. Sci. Comput.,2010,43(2):239-260.
[133]H. Han and Z. Huang. Tailored finite point method for steady-state reaction-diffusion equation [J]. Commun. Math. Sci.,2010,8:887-899.
[134]H. Han and Z. Huang. Tailored finite point method based on exponential bases for convection-diffusion-reaction equation [J]. Math. Comput.,2012, preprint.
[135]H. Han and Z. Huang. A tailored finite point method for the helmholtz equation with high wave numbers in heterogeneous medium [J]. J. Comput. Math.,26(5).
[136]P. W. Hsieh, Y. Shih, and S. Y. Yang. A tailored finite point method for solving steady MHD duct flow problems with boundary layers [J]. Commun. Comput. Phys.,2011,10(1):161-182.
[137]F. Bouchon and G. H. Peichl. A second-order immersed interface technique for an elliptic neumann problem [J]. Numer. Methods Partial Differential Equations,2007,23(2):400-420.
[138]J. Bedrossian, J. H. Von Brecht, S. Zhu, E. Sifakis, and J. M. Teran. A second order virtual node method for elliptic problems with interfaces and irregular domains [J]. J. Comput. Phys., 2010,229(18):6405-6426.
[139]J. L. Hellrung, L. Wang, E. Sifakis, and J. M. Teran. A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions [J]. J. Comput. Phys.,231(4):2015-2048.
[140]T. Chen and J. Strain. Piecewise-polynomial discretization and krylov-accelerated multigrid for elliptic interface problems [J]. J. Comput. Phys.,2008,227(16):7503-7542.
[141]K. Xia, M. Zhan, D. Wan, and G.W. Wei. Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients [J]. J. Comput. Phys.,231(4):1440-1461.
[142]W. C. Wang. A jump condition capturing finite difference scheme for elliptic interface problems [J]. SIAM J. Sci. Comput.,25(5).
[143]J. Jin and X. Wu. The singular finite element method for some elliptic boundary value problem with interface [J]. J. Comput. Appl. Math.,2004,170(1):197-216.
[144]S. Hou and X. D. Liu. A numerical method for solving variable coefficient elliptic equation with interfaces [J]. J. Comput. Phys.,2005,202(2):411-445.
[145]O. Iliev and S. Tiwari. A generalized (meshfree) finite difference discretization for elliptic interface problems [J]. Numer. Methods Appl.,2003, pages 488-497.
[146]孙志忠.偏微分方程数值解法[M].科学出版社,2005.
[147]D. W. Peaceman and H. H. Rachford Jr. The numerical solution of parabolic and elliptic differential equations [J]. J. Soc. Indust. Appl. Math.,1955,3(1):28-41.
[148]余德浩,汤华中.微分方程数值解法[M].科学出版社,2003.
[149]李荣华,冯果忱.微分方程数值解法[M].高等教育出版社,1999.
[150]D. L. Chopp. Some improvements of the fast marching method [J]. SIAM J. Sci. Comput., 2001,23:230-244.
[151]J. A. Sethian. Level set methods and fast marching methods:evolving interfaces in compu-tational geometry, fluid mechanics, computer vision, and materials science [M]. Cambridge University Press,1999.
[152]Y. H. R. Tsai, L. T. Cheng, S. Osher, and H. K. Zhao. Fast sweeping algorithms for a class of hamilton-jacobi equations [J]. SIAM J. Numer. Anal.,2004,41(2):673-694.
[153]H. Zhao and S. Osher.Visualization, analysis and shape reconstruction of unorganized data sets [J].2002.
[154]E. G. D'yakonov. Difference schemes with split operators for multidimensional unsteady prob-lems (English translation) [J]. USSR Comp. Math.,1963,3:581-607.
[155]N. Yanenko. Convergence of the method of splitting for the heat conduction equations with variable coefficients (English translation) [J]. USSR Comp. Math,1963,3:1094-1100.
[156]K. W. Morton and D. F. Mayers. Numerical solution of partial differential equations:an introduction [M]. Cambridge University Press,2005.
[157]Z. Li. A note on immersed interface method for three-dimensional elliptic equations [J]. Com-put. Math. Appl.,1996,31(3):9-17.
[158]W. F. Spotz. High-order compact finite difference schemes for computational mechanics [D]. University of Texas at Austin,1995.
[159]S. Karaa and J. Zhang. High order ADI method for solving unsteady convection-diffusion problems [J]. J. Comput. Phys.,2004,198(1):1-9.
[160]D. L. Brown, R. Cortez, and M. L. Minion. Accurate projection methods for the incompressible Navier-Stokes equations [J].J. Comput. Phys.,2001,168(2):464-499.
[161]R. T. Farouki and V. T. Rajan. On the numerical condition of polynomials in bernstein form [J]. Comput. Aided Geom. Design,1987,4(3):191-216.
[162]M. Idrees Bhatti and P. Bracken. Solutions of differential equations in a Bernstein polynomial basis [J]. J. Comput. Appl. Math.,2007,205(1):272-280.
[163]D. Bhatta. Use of modified Bernstein polynomials to solve KdV-Burgers equation numerically [J]. Appl. Math. Comp.,2008,206(1):457-464.
[164]B. N. Mandal and S. Bhattacharya. Numerical solution of some classes of integral equations using Bernstein polynomials [J]. Appl. Math. Comp.,2007,190(2):1707-1716.
[165]S. Bhattacharya and B. N. Mandal. Use of Bernstein polynomials in numerical solutions of Volterra integral equations [J]. Appl. Math. Sci.,2008,2(36):1773-1787.
[166]D. Wang and Z. Yang. Introduction to numerical approximation [M]. Higher Education Press, 1990.
[167]R. H. Goodman and R. Haberman. Chaotic scattering and the n-bounce resonance in solitary-wave interactions [J]. Phys. Rev. Lett.,2007,98(10):1041031-4.
[168]G. B. Jacobs and W. S. Don. A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks [J]. J. Comput. Phys.,2009, 228(5):1365-1379.
[169]Y. X. Li. Tango waves in a bidomain model of fertilization calcium waves [J]. Phys. D,2003, 186(1-2):27-49.
[170]X. Feng, Z. Li, and Z. Qiao. High order compact finite difference schemes for the helmholtz equation with discontinuous coefficients [J]. J. Comput. Math.,2011,29(3):324-340.
[2]Z. Li. An overview of the immersed interface method and its applications [J]. Taiwanese J. Math.,2003,7(1):1-49.
[3]A. McKenney, L. Greengard, and A. Mayo. A fast poisson solver for complex geometries [J]. J. Comput. Phys.,1995,118(2):348-355.
[4]R. J. LeVeque. Clawpack and amrclaw-software for high-resolution godunov methods [C]. Golden Colorado.
[5]S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations [J].J. Comput. Phys.,1988,79(l):12-49.
[6]L. Adams. A multigrid algorithm for immersed interface problems [C]. Citeseer,1996, pages 1-14.
[7]P. M. De Zeeuw. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver [J]. J. Comput. Appl. Math.,1990,33(1):1-27.
[8]C. S. Peskin. The immersed boundary method [J]. Acta Numer.,2002,11(1):479-517.
[9]M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to in-compressible two-phase flow [J]. J. Comput. Phys.,1994,114:146-159.
[10]Z. Li and K. Ito. The immersed interface method:numerical solutions of PDEs involving interfaces and irregular domains [M]. Soc. Ind. Appl. Math.,2006.
[11]M. Kumar and P. Joshi. Some numerical techniques for solving elliptic interface problems [J]. Numerical Methods for Partial Differential Equations,2010.
[12]J. B. Bell, C. N. Dawson, and G. R. Shubin. An unsplit, higher order godunov method for scalar conservation laws in multiple dimensions [J]. J. Comput. Phys.,1988,74(1):1-24.
[13]G. R. Shubin and J. B. Bell. An analysis of the grid orientation effect in numerical simulation of miscible displacement [J]. Comput. Methods Appl. Mech. Engrg.,1984,47(1-2):47-71.
[14]Z. Li. The Immersed Interface Method:A Numerical Approach for Partial Differential Equa-tions with Interfaces [D]. Seattle Washington, University of Washington,1994.
[15]C. S. Peskin. Flow patterns around heart valves:a numerical method [J]. J. Comput. Phys., 1972,10(2):252-271.
[16]C. S. Peskin. Numerical analysis of blood flow in the heart [J].J. Comput. Phys.,1977, 25(3):220-252.
[17]C. S. Peskin and D. M. McQueen. A three-dimensional computational method for blood flow in the heart. I. Immersed elastic fibers in a viscous incompressible fluid [J]. J. Comput. Phys., 1989,81(2):372-405.
[18]D. M. McQueen and C. S. Peskin. A three-dimensional computational method for blood flow in the heart. II. Contractile fibers [J]. J. Comput. Phys.,1989,82(2):289-297.
[19]R. P. Beyer Jr. A computational model of the cochlea using the immersed boundary method [J]. J. Comput. Phys.,1992,98(1):145-162.
[20]E. Givelberg. Modeling elastic shells immersed in fluid [J]. Commun. Pure Appl. Math.,2004, 57(3):283-309.
[21]A. L. Fogelson. A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting [J]. J. Comput. Phys.,1984,56(1):111-134.
[22]N. T. Wang and A. L. Fogelson. Computational Methods for Continuum Models of Platelet Aggregation [J]. J. Comput. Phys.,1999,151(2):649-675.
[23]L. J. Fauci and C. S. Peskin. A computational model of aquatic animal locomotion [J]. J. Comput. Phys.,1988,77(l):85-108.
[24]M. M. Hopkins and L. J. Fauci. A computational model of the collective fluid dynamics of motile micro-organisms [J]. J. Fluid Mech.,2002,455:149-174.
[25]R. Cortez, L. Fauci, N. Cowen, and R. Dillon. Simulation of swimming organisms:Coupling internal mechanics with external fluid dynamics [J]. Comput. Sci. Eng., IEEE,2005,6(3):38-45.
[26]L. Zhu and C. S. Peskin. Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method [J]. J. Comput. Phys.,2002,179(2):452-468.
[27]J. M. Stockie and S. I. Green. Simulating the Motion of Flexible Pulp Fibres Using the Im-mersed Boundary Method [J]. J. Comput. Phys.,1998,147(1):147-165.
[28]M. Uhlmann. An immersed boundary method with direct forcing for the simulation of partic-ulate flows [J]. J. Comput. Phys.,2005,209(2):448-476.
[29]L. A. Miller and C. S. Peskin. When vortices stick:an aerodynamic transition in tiny insect flight [J]. J. Exp. Biol.,2004,207(17):3073-3088.
[30]L. A. Miller and C. S. Peskin. A computational fluid dynamics of 'clap and fling' in the smallest insects [J]. J. Exp. Biol.,2005,208(2):195-212.
[31]R. J. Leveque and Z. Li. The immersed interface method for elliptic equations with discontin-uous coefficients and singular sources [J]. SIAM J. Numer. Anal.,1994,31(4):1019-1044.
[32]Z. Li and K. Ito. Maximum Principle Preserving Schemes for Interface Problems [J]. SIAM J. Sci. Comput.,2001,23:339-361.
[33]L. Adams and Z. Li. The immersed interface/multigrid methods for interface problems [J]. SIAM J. Sci. Comput.,2003,24(2):463-479.
[34]Z. Li. A fast iterative algorithm for elliptic interface problems [J]. SIAM J. Numer. Anal.,1998, 35(1):230-254.
[35]A. Mayo. The fast solution of Poisson's and the biharmonic equations on irregular regions [J]. SIAM J. Numer. Anal.,1984,21(2):285-299.
[36]A. Wiegmann and K. P. Bube. The explicit-jump immersed interface method:finite differ-ence methods for PDEs with piecewise smooth solutions [J]. SIAM J. Numer. Anal.,2000, 37(3):827-862.
[37]P. A. Berthelsen. A decomposed immersed interface method for variable coefficient ellip-tic equations with non-smooth and discontinuous solutions [J]. J. Comput. Phys.,2004, 197(1):364-386.
[38]P. Joshi and M. Kumar. Mathematical model and computer simulation of three dimensional thin film elliptic interface problems [J]. Comput. Math. Appl,2011,63(1):25-35.
[39]H. Huang and Z. Li. Convergence analysis of the immersed interface method [J]. IMA J. Numer. Anal.,1999,19(4):583-608.
[40]J. T. Beale and A. T. Layton. On the accuracy of finite difference methods for elliptic problems with interfaces [J]. Commun. Appl. Math. Comput. Sci.,2006,1(1):91-119.
[41]B.A. Vanderlei, M.M. Hopkins, and L.J. Fauci. Error estimation for immersed interface solu-tions [C].2007, pages 115-125.
[42]C. Zhang and R. J. LeVeque. The immersed interface method for acoustic wave equations with discontinuous coefficients [J]. Wave motion,1997,25(3):237-263.
[43]C. Zhang. Immersed interface methods for hyperbolic systems of partial differential equations with discontinuous coefficients [D]. University of Washington,1996.
[44]J. Piraux and B. Lombard. A new interface method for hyperbolic problems with discontinuous coefficients:one-dimensional acoustic example [J]. J. Comput. Phys.,2001,168(1):227-248.
[45]I. Brayanov, J. Kandilarov, and M. Koleva. Immersed interface difference schemes for a parabolic-elliptic interface problem [J]. Large-Scale Sci. Comput.,2008, pages 661-669.
[46]Z. Li and A. Mayo. ADI methods for heat equations with discontinuities along an arbitrary interface [C]. American Mathematical Society,1994,48:311-315.
[47]Z. Li and B. Soni. Fast and accurate numerical approaches for Stefan problems and crystal growth [J]. Numer. Heat TR. B-Fund.,1999,35(4):461-484.
[48]Z. Li. Immersed interface methods for moving interface problems [J]. Numer. Algorithms, 1997,14(4):269-293.
[49]A. Wiegmann and K. P. Bube. The immersed interface method for nonlinear differential equa-tions with discontinuous coefficients and singular sources [J]. SIAM J. Numer. Anal.,1998, 35(1):177-200.
[50]J. D. Kandilarov and L. G. Vulkov. The immersed interface method for two-dimensional heat-diffusion equations with singular own sources [J]. Appl. Numer. Math.,2007,57(5-7):486-497.
[51]Z. Li, D. Wang, and J. Zou. Theoretical and numerical analysis on a thermo-elastic system with discontinuities [J]. J. Comput. Appl. Math.,1998,92(1):37-58.
[52]I. Babuska. The finite element method for elliptic equations with discontinuous coefficients [J]. Computing,1970,5(3):207-213.
[53]J. H. Bramble and J. T. King. A finite element method for interface problems in domains with smooth boundaries and interfaces [J]. Adv. Comput. Math.,1996,6(1):109-138.
[54]J.W. Barrett and C.M. Elliott. Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces [J]. IMA J. Numer. Anal.,1987,7(3):283-300.
[55]C. Bernardi and R. Verfiirth. Adaptive finite element methods for elliptic equations with non-smooth coefficients [J]. Numer. Math.,2000,85(4):579-608.
[56]Z. Chen and S. Dai. On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients [J]. SIAM J. Sci. Comput.,2002,24(2):443-462.
[57]J. W. Barrett and C. M. Elliott. Finite element approximation of the Dirichlet problem using the boundary penalty method [J]. Numer. Math.,1986,49(4):343-366.
[58]R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids [J]. ESAIM-Math. Model. Numer. Anal.,2003,37(2):209-226.
[59]J. H. Bramble and J. T. King. A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries [J]. Math. Comp.,1994,63(207):1-18.
[60]Z. Chen and J. Zou. Finite element methods and their convergence for elliptic and parabolic interface problems [J]. Numer. Math.,1998,79(2):175-202.
[61]M. Feistauer and V. Sobotikova. Finite element approximation of nonlinear elliptic problems with discontinuous coefficients [J]. Comment. Math. Univ. Carolin.,1989,30(1):457-500.
[62]SP Marin. A finite element method for exterior interface problems [J]. Int. J. Math. Math. Sci., 1980,3:311-350.
[63]Z. Li. The immersed interface method using a finite element formulation [J]. Appl. Numer. Math.,1998,27(3):253-267.
[64]Z. Li, T. Lin, and X. Wu. New cartesian grid methods for interface problems using the finite element formulation [J]. Numer. Math.,2003,96(1):61-98.
[65]Y. Gong, B. Li, and Z. Li. Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions [J]. SIAM J. Numer. Anal,2008,46(1):472-495.
[66]Z. Li, T. Lin, Y. Lin, and RC Rogers. An immersed finite element space and its approximation capability [J]. Numer. Methods Partial Differential Equations,2004,20(3):338-367.
[67]L. Zhang, A. Gerstenberger, X. Wang, and W.K. Liu. Immersed finite element method [J]. Comput. Methods Appl. Mech. Engrg.,2004,193(21):2051-2067.
[68]X. He, T. Lin, and Y. Lin. Approximation capability of a bilinear immersed finite element space [J]. Numer. Methods Partial Differential Equations,2008,24(5):1265-1300.
[69]B. Camp, T. Lin, Y. Lin, and W. Sun. Quadratic immersed finite element spaces and their approximation capabilities [J]. Adv. Comput. Math.,2006,24(1):81-112.
[70]X. He, T. Lin, and Y. Lin. The convergence of the bilinear and linear immersed finite ele-ment solutions to interface problems [J]. Numer. Methods Partial Differential Equations,2012, 28(1):312-330.
[71]A. Hansbo and P. Hansbo. An unfitted finite element method, based on nitsche's method, for elliptic interface problems [J]. Comput. Methods Appl. Mech. Engrg.,2002,191(47):5537-5552.
[72]A. Loubenets, T. Ali, and M. Hanke. Highly accurate finite element method for one-dimensional elliptic interface problems [J]. Appl. Numer. Math.,2009,59(1):119-134.
[73]S. Adjerid and T. Lin. A p-th degree immersed finite element for boundary value problems with discontinuous coefficients [J]. Appl. Numer. Math.,2009,59(6):1303-1321.
[74]T. Lin, Y. Lin, W. W. Sun, and Z. Wang. Immersed finite element methods for 4th order differential equations [J]. J. Comput. Appl. Math.,2011,235(13):3953-3964.
[75]L. Mu, J. Wang, G. Wei, X. Ye, and S. Zhao. Weak galerkin methods for second order elliptic interface problems [J]. Arxiv preprint arXiv:1201.6438,2012.
[76]W. K. Liu, Y. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui, N. Patankar, Y. Zhang, C. Bajaj, J. Lee, et al. Immersed finite element method and its applications to biological systems [J]. Comput. Methods Appl. Mech. Engrg.,2006,195(13):1722-1749.
[77]J. Huang and J. Zou. Some new a priori estimates for second-order elliptic and parabolic interface problems [J]. J. Differential Equations,2002,184(2):570-586.
[78]W. K. Liu, D. W. Kim, and S. Tang. Mathematical foundations of the immersed finite element method [J]. Comput. Mech.,2007,39(3):211-222.
[79]L. T. Zhang and M. Gay. Immersed finite element method for fluid-structure interactions [J]. J. Fluid. Struct.,2007,23(6):839-857.
[80]S. H. Chou, D. Y. Kwak, and K. T. Wee. Optimal convergence analysis of an immersed interface finite element method [J]. Adv. Comput. Math.,2010,33(2):149-168.
[81]S. Wang and H. Chen. The optimal convergence analysis for an immersed finite element method [C]. IEEE,2011, pages 255-258.
[82]S. Adjerid and T. Lin. Higher-order immersed discontinuous galerkin methods [J]. Int. J. Inf. Syst. Sci.,2007,3(4):555-568.
[83]C. Engwer, P. Bastian, and S. P. Kuttanikkad. An unfitted discontinuous galerkin finite element method for pore scale simulations [C].2008,6126.
[84]P. Bastian, C. Engwer, J. Fahlke, and O. Ippisch. An unfitted discontinuous galerkin method for pore-scale simulations of solute transport [J]. Math. Comput. Simulation,2011, 81(10):2051-2061.
[85]X. He, T. Lin, and Y. Lin. Interior penalty bilinear ife discontinuous galerkin methods for elliptic equations with discontinuous coefficient [J]. J. Syst. Sci. Complex.,2010,23(3):467-483.
[86]F. Heimann, C. Engwer, O. Ippisch, and P. Bastian. An unfitted interior penalty discontinu-ous galerkin method for incompressible navier-stokes two-phase flow [J]. Internal J. Numer. Methods Fluids,2012.
[87]H. Wu and Y. Xiao. An unfitted hp-interface penalty finite element method for elliptic interface problems [J]. Arxiv preprint arXiv:1007.2893,2010.
[88]Y. R. Efendiev and X. H. Wu. Multiscale finite element for problems with highly oscillatory coefficients [J]. Numer. Math.,2002,90(3):459-486.
[89]CC Chu, I.G. Graham, and TY Hou. A new multiscale finite element method for high-contrast elliptic interface problems [J]. Math. Comp.,2010,79(272):1915-1955.
[90]R. R. Millward. A new adaptive multiscale finite element method with applications to high contrast interface problems [D]. University of Bath,2011.
[91]A. Liang, X. Jing, and X. Sun. Constructing spectral schemes of the immersed interface method via a global description of discontinuous functions [J]. J. Comput. Phys.,2008,227(18):8341-8366.
[92]Y. JIANG, A. LIANG, X. SUN, and X. JING. Multi-domain spectral immersed interface method for solving elliptic equation with a global description of discontinuous functions [J]. Chinese J. Aeronaut.,2012,25(3):297-310.
[93]M. N. Linnick and H. F. Fasel. A high-order immersed interface method for simulating un-steady incompressible flows on irregular domains [J].J. Comput. Phys.,2005,204(1):157-192.
[94]X. Zhong. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity [J]. J. Comput. Phys.,2007,225(1):1066-1099.
[95]I.T. Angelova and L.G. Vulkov. High-order finite difference schemes for elliptic problems with intersecting interfaces [J]. Appl. Math. Comput.,2007,187(2):824-843.
[96]X. D. Liu, R. P. Fedkiw, and M. Kang. A boundary condition capturing method for Poisson's equation on irregular domains [J]. J. Comput. Phys.,2000,160(l):151-178.
[97]X.D. Liu and T.C. Sideris. Convergence of the ghost fluid method for elliptic equations with interfaces [J]. Math. Comp.,2003,72(244):1731-1746.
[98]M. Kang, R. P. Fedkiw, and X. D. Liu. A boundary condition capturing method for multiphase incompressible flow [J]. J. Sci. Comput.,2000,15(3):323-360.
[99]D. Q. Nguyen, R. P. Fedkiw, and M. Kang. A boundary condition capturing method for incom-pressible flame discontinuities [J].J. Comput. Phys.,2001,172(1):71-98.
[100]D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle level set method for improved interface capturing [J]. J. Comput. Phys.,2002,183(1):83-116.
[101]F. Gibou, R. P. Fedkiw, L. T. Cheng, and M. Kang. A second-order-accurate symmetric discretization of the poisson equation on irregular domains [J]. J. Comput. Phys.,2002, 176(1):205-227.
[102]B. C. Shin and J. H. Jung. Spectral collocation and radial basis function methods for one-dimensional interface problems [J]. Appl. Numer. Math.,2011,61 (8):911-928.
[103]A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome. A conservative adaptive projection method for the variable density incompressible navier-stokes equations [J]. J. Comput. Phys.,1998,142(1):1-46.
[104]A. S. Almgren, J. B. Bell, P. Colella, and T. Marthaler. A cartesian grid projection method for the incompressible Euler equations in complex geometries [J]. SIAM J. Sci. Comput.,1997, 18(5):1289-1309.
[105]R. Ewing, O. Iliev, and R. Lazarov. A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients [J]. SIAM J. Sci. Comput.,2001,23(4):1335-1351.
[106]H. Johansen and P. Colella. A cartesian grid embedded boundary method for Poisson's equation on irregular domains [J]. J. Comput. Phys.,1998,147(1):60-85.
[107]R. E. Ewing, Z. Li, T. Lin, and Y. Lin. The immersed finite volume element methods for the elliptic interface problems [J]. Math. Comput. Simulation,1999,50(l):63-76.
[108]A. Naumovich. On a finite volume discretization of the three-dimensional biot poroelasticity system in multilayered domains [J]. Comput. Methods Appl. Math.,2006,6(3):306.
[109]XM He, T. Lin, and Y. Lin. A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficient [J]. Commun. Comput. Phys.,2010,6(1):185-202.
[110]M. Oevermann, C. Scharfenberg, and R. Klein. A sharp interface finite volume method for elliptic equations on cartesian grids [J]. J. Comput. Phys.,2009,228(14):5184-5206.
[111]A. Mayo. A decomposition finite difference method for the fourth order accurate solution of Poisson's equation on general regions [J]. Int. J. High Speed Comput.,1991,3:89-106.
[112]A. Mayo. The rapid evaluation of volume integrals of potential theory on general regions [J]. J. Comput. Phys.,1992,100(2):236-245.
[113]A. Greenbaum, L. Greengard, and A. Mayo. On the numerical solution of the biharmonic equation in the plane [J]. Phys. D,1992,60(1-4):216-225.
[114]A. Mayo. High order accurate particular solutions of the biharmonic equation on general regions [J]. Contemp. Math.,2003,323:233-244.
[115]A. Mayo and A. Greenbaum. Fast parallel iterative solution of poisson's and the biharmonic equations on irregular regions [J]. SIAM J. Sci. Statist. Comput.,1992,13(1):101-118.
[116]L. Greengard, M.C. Kropinski, and A. Mayo. Integral equation methods for Stokes flow and isotropic elasticity in the plane [J]. J. Comput. Phys.,1996,125(2):403-414.
[117]A. Mayo. Rapid, fourth order accurate solution of the steady Navier Stokes equations on general regions [J]. Dyn. Contin. Discrete Impuls. Syst.,2005,12:59-72.
[118]YC Zhou, S. Zhao, M. Feig, and GW Wei. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources [J]. J. Comput. Phys., 2006,213(1):1-30.
[119]SN Yu, Y. Xiang, and GW Wei. Matched interface and boundary (MIB) method for the vibra-tion analysis of plates [J]. Comm. Numer. Methods Engrg.,2009,25(9):923-950.
[120]S. Yu, Y. Zhou, and G. W. Wei. Matched interface and boundary (mib) method for elliptic problems with sharp-edged interfaces [J].J. Comput. Phys.,2007,224(2):729-756.
[121]S. Yu and G. W. Wei. Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities [J]. J. Comput. Phys.,2007,227(1):602-632.
[122]YC Zhou, J. Liu, and D.L. Harry. A matched interface and boundary method for solving multi-flow navier-stokes equations with applications to geodynamics [J]. Journal of Computational Physics,2011,231(1):223-242.
[123]S. Zhao and G. W. Wei. Matched interface and boundary (mib) for the implementation of boundary conditions in high-order central finite differences [J]. Internat. J. Numer. Methods Engrg.,2009,77(12):1690-1730.
[124]S. Zhao. High order matched interface and boundary methods for the helmholtz equation in media with arbitrarily curved interfaces [J]. J. Comput. Phys.,2010,229(9):3155-3170.
[125]K. Pan, Y. Tan, and H. Hu. An interpolation matched interface and boundary method for elliptic interface problems [J]. J. Comput. Appl. Math.,2010,234(l):73-94.
[126]I. Chern and Y. Shu. A coupling interface method for elliptic interface problems [J]. J. Comput. Phys.,2007,225(2):2138-2174.
[127]Y. C. Shu, C. Y. Kao, I. Chern, and C. C. Chang. Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients [J]. J. Comput. Phys.,2010, 229(24):9246-9268.
[128]J. Kandilarov. A coupling interface method for a nonlinear parabolic-elliptic problem [J]. Numer. Anal. Appl.,2009, pages 330-337.
[129]H. Han, Z. Huang, and R.B. Kellogg. A tailored finite point method for a singular perturbation problem on an unbounded domain [J]. J. Sci. Comput.,2008,36(2):243-261.
[130]H. Han and Z. Huang. Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions [J]. J. Sci. Comput.,2009,41(2):200-220.
[131]Z. Huang. Tailored finite point method for the interface problem [J]. Netw. Heterog. Media, 2009,4(1):91-106.
[132]Y. Shih, R.B. Kellogg, and P. Tsai. A tailored finite point method for convection-diffusion-reaction problems [J]. J. Sci. Comput.,2010,43(2):239-260.
[133]H. Han and Z. Huang. Tailored finite point method for steady-state reaction-diffusion equation [J]. Commun. Math. Sci.,2010,8:887-899.
[134]H. Han and Z. Huang. Tailored finite point method based on exponential bases for convection-diffusion-reaction equation [J]. Math. Comput.,2012, preprint.
[135]H. Han and Z. Huang. A tailored finite point method for the helmholtz equation with high wave numbers in heterogeneous medium [J]. J. Comput. Math.,26(5).
[136]P. W. Hsieh, Y. Shih, and S. Y. Yang. A tailored finite point method for solving steady MHD duct flow problems with boundary layers [J]. Commun. Comput. Phys.,2011,10(1):161-182.
[137]F. Bouchon and G. H. Peichl. A second-order immersed interface technique for an elliptic neumann problem [J]. Numer. Methods Partial Differential Equations,2007,23(2):400-420.
[138]J. Bedrossian, J. H. Von Brecht, S. Zhu, E. Sifakis, and J. M. Teran. A second order virtual node method for elliptic problems with interfaces and irregular domains [J]. J. Comput. Phys., 2010,229(18):6405-6426.
[139]J. L. Hellrung, L. Wang, E. Sifakis, and J. M. Teran. A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions [J]. J. Comput. Phys.,231(4):2015-2048.
[140]T. Chen and J. Strain. Piecewise-polynomial discretization and krylov-accelerated multigrid for elliptic interface problems [J]. J. Comput. Phys.,2008,227(16):7503-7542.
[141]K. Xia, M. Zhan, D. Wan, and G.W. Wei. Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients [J]. J. Comput. Phys.,231(4):1440-1461.
[142]W. C. Wang. A jump condition capturing finite difference scheme for elliptic interface problems [J]. SIAM J. Sci. Comput.,25(5).
[143]J. Jin and X. Wu. The singular finite element method for some elliptic boundary value problem with interface [J]. J. Comput. Appl. Math.,2004,170(1):197-216.
[144]S. Hou and X. D. Liu. A numerical method for solving variable coefficient elliptic equation with interfaces [J]. J. Comput. Phys.,2005,202(2):411-445.
[145]O. Iliev and S. Tiwari. A generalized (meshfree) finite difference discretization for elliptic interface problems [J]. Numer. Methods Appl.,2003, pages 488-497.
[146]孙志忠.偏微分方程数值解法[M].科学出版社,2005.
[147]D. W. Peaceman and H. H. Rachford Jr. The numerical solution of parabolic and elliptic differential equations [J]. J. Soc. Indust. Appl. Math.,1955,3(1):28-41.
[148]余德浩,汤华中.微分方程数值解法[M].科学出版社,2003.
[149]李荣华,冯果忱.微分方程数值解法[M].高等教育出版社,1999.
[150]D. L. Chopp. Some improvements of the fast marching method [J]. SIAM J. Sci. Comput., 2001,23:230-244.
[151]J. A. Sethian. Level set methods and fast marching methods:evolving interfaces in compu-tational geometry, fluid mechanics, computer vision, and materials science [M]. Cambridge University Press,1999.
[152]Y. H. R. Tsai, L. T. Cheng, S. Osher, and H. K. Zhao. Fast sweeping algorithms for a class of hamilton-jacobi equations [J]. SIAM J. Numer. Anal.,2004,41(2):673-694.
[153]H. Zhao and S. Osher.Visualization, analysis and shape reconstruction of unorganized data sets [J].2002.
[154]E. G. D'yakonov. Difference schemes with split operators for multidimensional unsteady prob-lems (English translation) [J]. USSR Comp. Math.,1963,3:581-607.
[155]N. Yanenko. Convergence of the method of splitting for the heat conduction equations with variable coefficients (English translation) [J]. USSR Comp. Math,1963,3:1094-1100.
[156]K. W. Morton and D. F. Mayers. Numerical solution of partial differential equations:an introduction [M]. Cambridge University Press,2005.
[157]Z. Li. A note on immersed interface method for three-dimensional elliptic equations [J]. Com-put. Math. Appl.,1996,31(3):9-17.
[158]W. F. Spotz. High-order compact finite difference schemes for computational mechanics [D]. University of Texas at Austin,1995.
[159]S. Karaa and J. Zhang. High order ADI method for solving unsteady convection-diffusion problems [J]. J. Comput. Phys.,2004,198(1):1-9.
[160]D. L. Brown, R. Cortez, and M. L. Minion. Accurate projection methods for the incompressible Navier-Stokes equations [J].J. Comput. Phys.,2001,168(2):464-499.
[161]R. T. Farouki and V. T. Rajan. On the numerical condition of polynomials in bernstein form [J]. Comput. Aided Geom. Design,1987,4(3):191-216.
[162]M. Idrees Bhatti and P. Bracken. Solutions of differential equations in a Bernstein polynomial basis [J]. J. Comput. Appl. Math.,2007,205(1):272-280.
[163]D. Bhatta. Use of modified Bernstein polynomials to solve KdV-Burgers equation numerically [J]. Appl. Math. Comp.,2008,206(1):457-464.
[164]B. N. Mandal and S. Bhattacharya. Numerical solution of some classes of integral equations using Bernstein polynomials [J]. Appl. Math. Comp.,2007,190(2):1707-1716.
[165]S. Bhattacharya and B. N. Mandal. Use of Bernstein polynomials in numerical solutions of Volterra integral equations [J]. Appl. Math. Sci.,2008,2(36):1773-1787.
[166]D. Wang and Z. Yang. Introduction to numerical approximation [M]. Higher Education Press, 1990.
[167]R. H. Goodman and R. Haberman. Chaotic scattering and the n-bounce resonance in solitary-wave interactions [J]. Phys. Rev. Lett.,2007,98(10):1041031-4.
[168]G. B. Jacobs and W. S. Don. A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks [J]. J. Comput. Phys.,2009, 228(5):1365-1379.
[169]Y. X. Li. Tango waves in a bidomain model of fertilization calcium waves [J]. Phys. D,2003, 186(1-2):27-49.
[170]X. Feng, Z. Li, and Z. Qiao. High order compact finite difference schemes for the helmholtz equation with discontinuous coefficients [J]. J. Comput. Math.,2011,29(3):324-340.