PUBLICATIONS
Zhiqiang Cai


PUBLICATIONS

  1. (with C. He and S. Zhang) Residual-based a posteriori error estimate forinterface problems: nonconforming linear elements, Math. Comp., to appear.
  2. (with B. Chen) Least squares method for the Oseen equation, Numer. Methods Part. Diff. Eqs. , to appear.
  3. (with S. Cao) A recovery-based a posteriori error estimator for H(curl) interface problems, Comput. Methods in Appl. Mech. Eng., 296 (2015), 169-195.
  4. (with R. Falgout and S. Zhang) Div first-order system LL* (FOSLL*) for second-order elliptic partial differential equations, SIAM J. Numer. Anal., 53:1 (2015), 405-420.
  5. (with V. Carey, J. Ku, and E.-J. Park) Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L2 norm, Comput. Math. Appl., 70 (2015), 648-659.
  6. (with L. Song and Y. Hou) Recovery-based error estimator for stabilized finite element methods for the Stokes equations, Comput. Methods in Appl. Mech. Eng., 272 (2014), 1-16.
  7. (with S. Zhang) Robust equilibrated residual error estimator for diffusion problems: Conforming elements, SIAM J. Numer. Anal., 50:1 (2012), 151-170.
  8. (with S. Zhang) Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation, Math. Comp., 81:280 (2012), 1903-1927.
  9. (with J. Ku) Goal-oriendted local a posteriori error estimator for H(div) least-squares finite element method, SIAM J. Numer. Anal., 49:6 (2011), 2564-2575.
  10. (with X. Ye and S. Zhang) Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations, SIAM J. Numer. Anal., 49:5 (2011), 1761-1787, CORRECTION.
  11. (with S. Zhang) Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps, Numer. Methods Part. Diff. Eqs. , 28:2 (2012), 476-491 (published online 10/25/2010).
  12. (with S. Zhang) Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations, SIAM J. Numer. Anal., 48:2 (2010), 578-602.
  13. (with C. Wang and S. Zhang) Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations, SIAM J. Numer. Anal., 48:1 (2010), 79-94.
  14. (with S. Zhang) Recovery-based error estimators for interface problems: mixed and nonconforming finite elements, SIAM J. Numer. Anal., 48:1 (2010), 30-52.
  15. (with Ku) Optimal error estimates for the div least-squares method with data f in L2 and application to nonlinear problems, SIAM J. Numer. Anal., 47:6 (2010), 4098-4111.
  16. (with Y. Wang) Pseudostress-velocity formulation for incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 63:3 (2010), 341-356.
  17. (with C. Tong, P. S. Vassilevski, and C. Wang) Mixed finite element method for Stokes equations based on pseudostress-velocityformulation , Numer. Methods Part. Diff. Eqs. , 26:4 (2010), 957-978.
  18. (with F. Attia and G. Starke) First-order system least squares for the Signorini contact problem in linear elasticity, SIAM J. Numer. Anal., 47:4 (2009), 3027-3043.
  19. (with C. Westphal) An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type, J. Non-Newt. Fluid Mech., 159:1-3 (2009), 72-80.
  20. (with S. Zhang) Recovery-based error estimators for interface problems: conforming linear elements, SIAM J. Numer. Anal., 47:3 (2009), 2132-2156.
  21. (with Y. Wang) An error estimate for two-dimensional Stokes driven cavity problem, Math. Comp., 78:266 (2009), 771-787.
  22. (with S. Zhang) Recovery-based error estimators for interface problems: mixed and nonconforming finite elements (extended version), manuscript, 2008.
  23. (with C. Westphal) A weighted H(div) least-squares method for second-order elliptic equations, SIAM J. Numer. Anal., 46:3 (2008), 1640-1651.
  24. (with Y. Wang) A multigrid method for the pseudostress formulation of the Stokes problems, SIAM J. Sci. Compt., 29:5 (2007), 2078-2095.
  25. (with S. Kim, J. Pyo, and S. Kong) A finite element method using singular functions: interface problems , Hokkaido Math. J. , 36 (2007), 815-836.
  26. (with J. Ku) The L2 norm error estimates for the Div least-squares method, SIAM J. Numer. Anal., 44:4 (2006), 1721-1734.
  27. (with S. Kim, S. Kim, and S. Kong) A finite element method using singular functions for Poisson equations: mixed boundary conditions , Comput. Methods Appl. Mech. Engrg., 195:19-22 (2006), 2635-2648.
  28. (with X. Ye) A mixed nonconforming finite element for linear elasticity , Numer. Methods Part. Diff. Eqs. , 21:6 (2005), 1043-1051.
  29. (with J. Korsawe and G. Starke) Adaptive least squares mixed finite element computations for the stress-displacement formulation of linear elasticity, Numer. Methods Part. Diff. Eqs. , 21:1 (2005), 132-148.
  30. (with B. Lee and P. Wang) Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems, SIAM J. Numer. Anal., 42:2 (2004), 843-859.
  31. (with G. Starke) Least-squares methods for linear elasticity, SIAM J. Numer. Anal., 42:2 (2004), 826-842.
  32. (with G. Starke) First-order system least squares for the stress-displacement formulation: linear elasticity, SIAM J. Numer. Anal., 41:2 (2003), 715-730.
  33. (with R. Parashkevov, T. Russell, J. Wilson, and X. Ye) Domain decomposition for a mixed finite element method in three dimensions, SIAM J. Numer. Anal., 41:1 (2003), 181-194.
  34. (with J. Douglas, Jr. and M. Park) Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Advan. Comp. Math., 19:1-3 (2003), 3-33.
  35. (with S. Kim and G. Woo) A finite element method using singular functions for the Poisson equation: crack singularities, Numer. Lin. Alg. Appl., 9:6-7 (2002), 445-455.
  36. (with B.-C. Shin) The discrete first-order system least squares: the second-order elliptic boundary value problem, SIAM J. Numer. Anal., 40:1 (2002), 307-318.
  37. (with T. Manteuffel, S. McCormick, and J. Ruge) First-order system LL* (FOSLL*): scalar elliptic partial differential equations, SIAM J. Numer. Anal., 39:4 (2001), 1418-1445.
  38. (with S. Kim and B.-C. Shin) Solution methods for the Poisson equation with corner singularities: numerical results, SIAM J. Sci. Comput., 23:2 (2001), 672-682.
  39. (with S. Kim) A finite element method using singular functions for the Poisson equation: corner singularities, SIAM J. Numer. Anal., 39:1 (2001), 286-299.
  40. Least squares for the perturbed Stokes equations and the Reissner-Mindlin plate, SIAM J. Numer. Anal., 38:5 (2000), 1561--1581.
  41. (with J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye) Nonconforming quadrilateral finite elements: a correction, Calcolo, 37 (2000), 253--254.
  42. (with C.-O. Lee, T. Manteuffel, and S. McCormick) First-order system least squares for the Stokes and linear elasticity equations: further results, SIAM J. Sci. Comput., 21:5 (2000), 1728--1739.
  43. (with C.-O. Lee, T. Manteuffel, and S. McCormick) First-order system least squares for linear elasticity: numerical results, SIAM J. Sci. Comput., 21:5 (2000), 1706--1727.
  44. (with X. Ye) Least-squares finite element approximations for compressible Stokes equation, Numer. Methods Part. Diff. Eqs, 16 (2000), 62-70.
  45. (with J. Douglas, Jr. and X. Ye) A stable nonconforming quadrilateral finite element method for the Stokes and Navier-Stokes equations, Calcolo, 36 (1999), 215-232.
  46. (with X. Ye and H. Zhang) Least-squares finite element approximations for the Reissner-Mindlin plate, Numer. Lin. Alg. Appl., 6:6 (1999), 479-496.
  47. (with T. Hou and X. Wu) Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68:227 (1999), 913-943.
  48. (with J. Douglas, Jr.) Stabilized finite element methods with fast iterative solution algorithms for the Stokes problem, Comput. Methods Appl. Mech. Engrg., 166 (1998), 115-129.
  49. (with P. Bochev, T. Manteuffel, and S. McCormick) Analysis of velocity-flux least-squares principles for Navier-Stokes equations: Part I, SIAM J. Numer. Anal., 35:3 (1998), 990-1009.
  50. (with R. Parashkevov, T. Russell, and X. Ye) Overlapping domain decomposition for a mixed finite element method in three dimensions, In P. Bjorstad, M. Espedal, and D. Keyes (eds.) the 9th International Conference on Domain Decomposition Methods, Bergen, Norway, 1998, 188-196.
  51. (with T. Manteuffel, S. McCormick, and S. Parter) First-order system least squares for planar linear elasticity: pure traction problem, SIAM J. Numer. Anal., 35:1 (1998), 320--335.
  52. (with T. Manteuffel and S. McCormick) First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal., 34:5 (1997), 1727--1741.
  53. (with J. Douglas, Jr.) An analytic basis for multigrid methods for stabilized finite element methods for the Stokes problem, in Computational Science for the 21st Century, M-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. P\'{e}riaux, M.F. Wheeler, Eds, Wiley, 1997, pp.~113--118.
  54. (with J. Jones, S. McCormick, and T. Russell) Control-volume mixed finite element methods, Computational Geosciences, 1 (1997), 289-315.
  55. (with T. Manteuffel and S. McCormick) First-order system least squares for second-order partial differential equations: Part II, SIAM J. Numer. Anal., 34:2 (1997), 425--454.
  56. (with J. Mandel and S. McCormick) Multigrid methods for nearly singular equations and eigenvalue problems, SIAM J. Numer. Anal., 34:1 (1997), 178--200.
  57. (with G. Lai) Convergence analysis of multigird method for nonsymmetric and indefinite elliptic problems, Numer. Lin. Alg. Appl., 3:3 (1996), 205--220.
  58. (with F. Le Gland and H. Zhang) An adaptive local refinement techniques for nonlinear filtering, Publication Interne No. 954, IRISA, France, October 1995.
  59. (with T. Manteuffel and S. McCormick) First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity, ETNA, 3 (1995), 150--159.
  60. (with P. Bochev, T. Manteuffel, and S. McCormick) First-order system least squares for the Navier-Stokes equations, Proceedings of the Seventh Copper Moutain Conferences on Multigrid Methods, 1995, NASA Conference Publication 3339, 41-55.
  61. (with R. Lazorov, T. Manteuffel, and S. McCormick) First-order system least squares for second-order partial differential equations: Part I, SIAM J. Numer. Anal., 31:6 (1994) 1785--1799.
  62. (with C. Goldstein and J. Pasciak) Multilevel iteration for mixed finite element systems with penalty, SIAM J. Sci. Comput., 14:5 (1993), 1072--1088.
  63. Norm estimates of product operators with application to domain decomposition, Appl. Math. Comp., 53 (1993), 251--276.
  64. (with S. McCormick) Schwarz alternating procedure for elliptic problems discretized by least squares mixed finite elements, manuscript, 1993.
  65. (with Weinan E) Hierarchical method for elliptic problems using wavelet, Comm. Appl. Numer. Methods, 8 (1992), 819--825.
  66. On the finite volume element method, Numer. Math., 58 (1991), 713--735.
  67. (with J. Mandel and S. McCormick) The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28:2 (1991), 392--402.
  68. A Theoretical Foundation for the Finite Volume Element Method, Ph.D Thesis, University of Colorado at Denver, May, 1990.
  69. (with S. McCormick) On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J. Numer. Anal., 27:3 (1990), 636--655.
  70. (with S. McCormick) Computational complexity of the Schwarz alternating procedure, Int. J. High Speed Computing, 1 (1989), 1--28.
  71. (with B. Y. Gao) Error Estimates for the Schwarz alternating procedure on L-shaped regions, Appl. Math. Comp., 28 (1988), 39--46.
  72. Multigrid method with correction procedure, J. Comput. Math., 5:4 (1987), 336--341.
  73. (with N.C. Wang) The revised MGE method, Numer. Math. J. Chinese Univ., 9:3 (1987), 227--234.
  74. (with N.C. Wang and L.S. Kang) The MGE method for solving elliptic boundary value problems, Math. Numer. Sinica, 8:1 (1986), 82--89.
  75. Multigrid iteration with defect correction, J. Huazhong Univ. Sci. Tech., 14 (1986), 13--16.
  76. Splitting extrapolation and its parallel algorithm for nonlinear elliptic boundary value problems, Computer Sci. Engin., 4 (1985), 1--5.
  77. Optimal error bound for a class of comparatively generalized Hermite interpolation, J. Engin. Math., 2 (1985), 152--154.