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Department of Mathematics

Dr. Zhiqiang Cai

Dr. Zhiqiang Cai Professor of Mathematics, Interim Director of Center for Computational and Applied Mathematics
  • 765-49-41921
  • MATH 412
caiz@purdue.edu

Personal Website

Research Interest(s):
numerical analysis, applied mathematics

Publications listed in MathSciNet

  • [1] Zhiqiang Cai, Seokchan Kim, Sangdong Kim, and Sooryun Kong. A finite element method using singular functions for Poisson equations: mixed boundary conditions. Comput. Methods Appl. Mech. Engrg., 195(19-22):2635-2648, 2006.
  • [2] Zhiqiang Cai and Xiu Ye. A mixed nonconforming finite element for linear elasticity. Numer. Methods Partial Differential Equations, 21(6):1043-1051, 2005.
  • [3] Zhiqiang Cai, Johannes Korsawe, and Gerhard Starke. An adaptive least squares mixed finite element method for the stress-displacement formulation of linear elasticity. Numer. Methods Partial Differential Equations, 21(1):132-148, 2005.
  • [4] Zhiqiang Cai, Barry Lee, and Ping Wang. Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal., 42(2):843-859 (electronic), 2004.
  • [5] Zhiqiang Cai and Gerhard Starke. Least-squares methods for linear elasticity. SIAM J. Numer. Anal., 42(2):826-842 (electronic), 2004.
  • [6] Zhiqiang Cai and Gerhard Starke. First-order system least squares for the stress-displacement formulation: linear elasticity. SIAM J. Numer. Anal., 41(2):715-730 (electronic), 2003.
  • [7] Zhiqiang Cai, Jim Douglas, Jr., and Moongyu Park. Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Adv. Comput. Math., 19(1-3):3-33, 2003. Challenges in computational mathematics (Pohang, 2001).
  • [8] Z. Cai, R. R. Parashkevov, T. F. Russell, J. D. Wilson, and X. Ye. Domain decomposition for a mixed finite element method in three dimensions. SIAM J. Numer. Anal., 41(1):181-194 (electronic), 2003.
  • [9] Zhiqiang Cai, Seokchan Kim, and Gyungsoo Woo. A finite element method using singular functions for the Poisson equation: crack singularities. Numer. Linear Algebra Appl., 9(6-7):445-455, 2002. Preconditioned robust iterative solution methods, PRISM '01 (Nijmegen).
  • [10] Zhiqiang Cai and Byeong Chun Shin. The discrete first-order system least squares: the second-order elliptic boundary value problem. SIAM J. Numer. Anal., 40(1):307-318 (electronic), 2002.
  • [11] Z. Cai, T. A. Manteuffel, S. F. McCormick, and J. Ruge. First-order system LL* (FOSLL*): scalar elliptic partial differential equations. SIAM J. Numer. Anal., 39(4):1418-1445 (electronic), 2001.
  • [12] Zhiqiang Cai, Seokchan Kim, and Byeong-Chun Shin. Solution methods for the Poisson equation with corner singularities: numerical results. SIAM J. Sci. Comput., 23(2):672-682 (electronic), 2001. Copper Mountain Conference (2000).
  • [13] Zhiqiang Cai and Seokchan Kim. A finite element method using singular functions for the Poisson equation: corner singularities. SIAM J. Numer. Anal., 39(1):286-299 (electronic), 2001.
  • [14] Zhiqiang Cai. Least squares for the perturbed Stokes equations and the Reissner-Mindlin plate. SIAM J. Numer. Anal., 38(5):1561-1581 (electronic), 2000.
  • [15] Z. Cai, J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye. Nonconforming quadrilateral finite elements: a correction. Calcolo, 37(4):253-254, 2000.
  • [16] Z. Cai, C.-O. Lee, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for the Stokes and linear elasticity equations: further results. SIAM J. Sci. Comput., 21(5):1728-1739 (electronic), 2000. Iterative methods for solving systems of algebraic equations (Copper Mountain, CO, 1998).
  • [17] Z. Cai, C.-O. Lee, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for linear elasticity: numerical results. SIAM J. Sci. Comput., 21(5):1706-1727 (electronic), 2000. Iterative methods for solving systems of algebraic equations (Copper Mountain, CO, 1998).
  • [18] Zhiqiang Cai and Xiu Ye. A least-squares finite element approximation for the compressible Stokes equations. Numer. Methods Partial Differential Equations, 16(1):62-70, 2000.
  • [19] Zhiqiang Cai, Jim Douglas, Jr., and Xiu Ye. A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo, 36(4):215-232, 1999.
  • [20] Zhiqiang Cai, Xiu Ye, and Huilong Zhang. Least-squares finite element approximations for the Reissner-Mindlin plate. Numer. Linear Algebra Appl., 6(6):479-496, 1999. Iterative solution methods for the elasticity equations in mechanics and biomechanics, IMMB'98, Part 1 (Nijmegen).
  • [21] Thomas Y. Hou, Xiao-Hui Wu, and Zhiqiang Cai. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp., 68(227):913-943, 1999.
  • [22] Zhiqiang Cai and Jim Douglas, Jr. Stabilized finite element methods with fast iterative solution algorithms for the Stokes problem. Comput. Methods Appl. Mech. Engrg., 166(1-2):115-129, 1998.
  • [23] P. Bochev, Z. Cai, T. A. Manteuffel, and S. F. McCormick. Analysis of velocity-flux first-order system least-squares principles for the Navier-Stokes equations. I. SIAM J. Numer. Anal., 35(3):990-1009 (electronic), 1998.
  • [24] Zhiqiang Cai, Thomas A. Manteuffel, Stephen F. McCormick, and Seymour V. Parter. First-order system least squares (FOSLS) for planar linear elasticity: pure traction problem. SIAM J. Numer. Anal., 35(1):320-335 (electronic), 1998.
  • [25] Z. Cai, J. E. Jones, S. F. McCormick, and T. F. Russell. Control-volume mixed finite element methods. Comput. Geosci., 1(3-4):289-315 (1998), 1997.
  • [26] Z. Cai, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for the Stokes equations, with application to linear elasticity. SIAM J. Numer. Anal., 34(5):1727-1741, 1997.
  • [27] Zhiqiang Cai, Jan Mandel, and Steve McCormick. Multigrid methods for nearly singular linear equations and eigenvalue problems. SIAM J. Numer. Anal., 34(1):178-200, 1997.
  • [28] Zhiqiang Cai, Thomas A. Manteuffel, and Stephen F. McCormick. First-order system least squares for second-order partial differential equations. II. SIAM J. Numer. Anal., 34(2):425-454, 1997.
  • [29] Zhiqiang Cai and Chen-Yao G. Lai. Convergence estimates of multilevel additive and multiplicative algorithms for non-symmetric and indefinite problems. Numer. Linear Algebra Appl., 3(3):205-220, 1996.
  • [30] Zhiqiang Cai, Thomas A. Manteuffel, and Stephen F. McCormick. First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity. Electron. Trans. Numer. Anal., 3(Dec.):150-159 (electronic), 1995.
  • [31] Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal., 31(6):1785-1799, 1994.
  • [32] Zhi Qiang Cai, Charles I. Goldstein, and Joseph E. Pasciak. Multilevel iteration for mixed finite element systems with penalty. SIAM J. Sci. Comput., 14(5):1072-1088, 1993.
  • [33] Zhi Qiang Cai. Norm estimates of product operators with application to domain decomposition. Appl. Math. Comput., 53(2-3):251-276, 1993.
  • [34] Zhi Qiang Cai and Weinan E. Hierarchical method for elliptic problems using wavelet. Comm. Appl. Numer. Methods, 8(11):819-825, 1992.
  • [35] Zhi Qiang Cai. On the finite volume element method. Numer. Math., 58(7):713-735, 1991.
  • [36] Zhi Qiang Cai, Jan Mandel, and Steve McCormick. The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal., 28(2):392-402, 1991.
  • [37] Zhi Qiang Cai and Steve McCormick. On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal., 27(3):636-655, 1990.
  • [38] Zhi Qiang Cai and Benju Guo. Error estimates for a Schwarz alternating procedure on L-shaped regions. Appl. Math. Comput., 28(1):39-46, 1988.
  • [39] Zhi Qiang Cai. The multigrid method with correction procedure. J. Comput. Math., 5(4):336-341, 1987.
  • [40] Neng Chao Wang and Zhi Qiang Cai. The revised MGE method. Numer. Math. J. Chinese Univ., 9(3):227-234, 1987.
  • [41] Zhi Qiang Cai, Neng Chao Wang, and Li Shan Kang. The MGE method for elliptic boundary value problems. Math. Numer. Sinica, 8(1):82-89, 1986.

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