Edit Profile (opens in new tab) Cai, Zhiqiang Co-Author Distance Author ID: cai.zhiqiang Published as: Cai, Zhiqiang; Cai, Z.; Cai, Zhi-qiang; Chai, Zhiqiang more...less Documents Indexed: 118 Publications since 1986, including 9 Additional arXiv Preprints 1 Contribution as Editor Co-Authors: 85 Co-Authors with 102 Joint Publications 2,750 Co-Co-Authors all top 5 Co-Authors 6 single-authored 19 Zhang, Shun 14 McCormick, Steve Fahrney 10 Liu, Min 9 Manteuffel, Thomas A. 7 Ye, Xiu 6 Chen, Jingshuang 6 He, Cuiyu 6 Kim, Seokchan 5 Douglas, Jim jun. 5 Ku, Jaeun 4 Cai, Difeng 4 Si, Shubin 4 Starke, Gerhard 3 Cao, Shuhao 3 Dui, Hongyan 3 Park, Moongyu 3 Sun, Shudong 3 Wang, Yanqiu 2 Chen, Binghe 2 Choi, Junpyo 2 Dawson, Darren M. 2 de Queiroz, Marcio S. 2 Du, Shaohong 2 Falgout, Rob 2 Hao, Zhaopeng 2 Lee, Chang-Ock 2 Mandel, Jan 2 Russell, Thomas F. 2 Shin, Byeong Chun 2 Wang, Chunbo 2 Wang, Nengchao 2 Westphal, Chad R. 2 Zhang, Yingfeng 1 Aikawa, K. 1 Attia, Frank S. 1 Baier, Stephan 1 Bank, Randolph E. 1 Baranov, M. A. 1 Bertrand, Fleurianne 1 Bochev, Pavel B. 1 Carey, Varis 1 Chomaz, L. 1 Du, Ding-Zhu 1 Duan, Dongli 1 E, Weinan 1 Falgout, Robert D. 1 Ferlaino, F. 1 Goldstein, Charles I. 1 Guo, Ben-Yu 1 Hejnal, Brooke 1 Hou, Thomas Yizhao 1 Hou, Yanren 1 Jones, Jim E. 1 Kang, Lishan 1 Kim, Sangdong 1 Kong, Sooryoun 1 Kong, Sooryun 1 Korsawe, Johannes 1 Lai, Chen-Yao G. 1 Lazarov, Raytcho D. 1 Lee, Barry 1 Lee, Hyung-Chun 1 Lin, Guang 1 Liu, Baoliang 1 Liu, Xinyu 1 Mark, M. J. 1 Meng, Xueyu 1 Parashkevov, Rossen R. 1 Park, Eun-Jae 1 Park, Eunyoung 1 Parter, Seymour V. 1 Pasciak, Joseph E. 1 Petter, D. 1 Pyo, Jae-Hong 1 Qi, Huan 1 Ramani, Karthik 1 Ruge, J. 1 Santos, Juan Enrique 1 Saygin, Can 1 Shang, Lijun 1 Sheen, Dongwoo 1 Song, Lina 1 Song, Xiaoyu 1 Thai, My T. 1 Vassilevski, Panayot Spirov 1 Verfürth, Rüdiger 1 Wang, Junbo 1 Wang, Ning 1 Wang, Ping 1 Wilson, John David 1 Woo, Gyungsoo 1 Wu, Cang 1 Wu, Xiaohui 1 Xia, Jun 1 Zhang, Huilong 1 Zhang, Shuai 1 Zhang, Yang 1 Zhao, Xibin 1 Zoller, Peter all top 5 Serials 30 SIAM Journal on Numerical Analysis 8 Computer Methods in Applied Mechanics and Engineering 6 Numerical Methods for Partial Differential Equations 5 Computers & Mathematics with Applications 5 SIAM Journal on Scientific Computing 4 Mathematics of Computation 3 Journal of Computational Physics 3 Applied Mathematics and Computation 3 Applied Numerical Mathematics 3 Journal of Scientific Computing 3 Communications in Statistics. Theory and Methods 3 Numerical Linear Algebra with Applications 2 Calcolo 2 IEEE Transactions on Automatic Control 2 Journal of Computational and Applied Mathematics 2 Numerische Mathematik 1 Acta Mechanica 1 Classical and Quantum Gravity 1 International Journal for Numerical Methods in Fluids 1 Hokkaido Mathematical Journal 1 Numerical Mathematics 1 Mathematica Numerica Sinica 1 Journal of Computational Mathematics 1 Communications in Applied Numerical Methods 1 International Journal of High Speed Computing 1 Applied Mathematical Modelling 1 Journal of Non-Newtonian Fluid Mechanics 1 ETNA. Electronic Transactions on Numerical Analysis 1 Advances in Computational Mathematics 1 Mathematical Problems in Engineering 1 Optimization Methods & Software 1 Journal of Ningxia University. Natural Science Edition 1 Computational Geosciences 1 Computational Methods in Applied Mathematics 1 Communications in Mathematical Sciences 1 Journal of Computational Mathematics and Optimization 1 Oberwolfach Reports 1 Science 1 East Asian Journal on Applied Mathematics all top 5 Fields 85 Numerical analysis (65-XX) 50 Partial differential equations (35-XX) 25 Fluid mechanics (76-XX) 18 Mechanics of deformable solids (74-XX) 5 Computer science (68-XX) 5 Operations research, mathematical programming (90-XX) 4 Probability theory and stochastic processes (60-XX) 4 Statistics (62-XX) 2 Approximations and expansions (41-XX) 2 Biology and other natural sciences (92-XX) 2 Systems theory; control (93-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 Real functions (26-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Difference and functional equations (39-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Optics, electromagnetic theory (78-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Relativity and gravitational theory (83-XX) 1 Astronomy and astrophysics (85-XX) 1 Geophysics (86-XX) Publications by Year all cited Publications top 5 cited Publications Citations contained in zbMATH Open 92 Publications have been cited 2,182 times in 1,357 Documents Cited by ▼ Year ▼ On the finite volume element method. Zbl 0731.65093 Cai, Zhiqiang 233 1991 Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Zbl 0922.65071 Hou, Thomas Y.; Wu, Xiao-Hui; Cai, Zhiqiang 215 1999 First-order system least squares for second-order partial differential equations. I. Zbl 0813.65119 Cai, Z.; Lazarov, R.; Manteuffel, T. A.; McCormick, S. F. 144 1994 The finite volume element method for diffusion equations on general triangulations. Zbl 0729.65086 Cai, Zhiqiang; Mandel, Jan; McCormick, Steve 123 1991 On the accuracy of the finite volume element method for diffusion equations on composite grids. Zbl 0707.65073 Cai, Zhiqiang; McCormick, Steve 91 1990 First-order system least squares for second-order partial differential equations: Part II. Zbl 0912.65089 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F. 86 1997 Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. Zbl 1232.65142 Cai, Zhiqiang; Ye, Xiu; Zhang, Shun 79 2011 First-order system least squares for the Stokes equations, with application to linear elasticity. Zbl 0901.76052 Cai, Z.; Manteuffel, T. A.; McCormick, S. F. 76 1997 A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Zbl 0947.76047 Cai, Zhiqiang; Douglas, Jim jun.; Ye, Xiu 57 1999 Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. Zbl 1159.76347 Cai, Zhiqiang; Lee, Barry; Wang, Ping 57 2004 Least-squares methods for linear elasticity. Zbl 1159.74419 Cai, Zhiqiang; Starke, Gerhard 56 2004 Recovery-based error estimator for interface problems: Conforming linear elements. Zbl 1204.65129 Cai, Zhiqiang; Zhang, Shun 53 2009 Control-volume mixed finite element methods. Zbl 0941.76050 Cai, Z.; Jones, J. E.; McCormick, S. F.; Russell, T. F. 50 1997 Analysis of velocity-flux first-order system least-squares principles for the Navier-Stokes equations. I. Zbl 0910.76032 Bochev, P.; Cai, Z.; Manteuffel, T. A.; McCormick, S. F. 48 1998 A finite element method using singular functions for the Poisson equation: Corner singularities. Zbl 0992.65122 Cai, Zhiqiang; Kim, Seokchan 43 2001 Mixed finite element methods for incompressible flow: stationary Stokes equations. Zbl 1267.76059 Cai, Zhiqiang; Tong, Charles; Vassilevski, Panayot S.; Wang, Chunbo 43 2010 Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Zbl 1020.65087 Cai, Zhiqiang; Douglas, Jim jun.; Park, Moongyu 40 2003 Recovery-based error estimators for interface problems: mixed and nonconforming finite elements. Zbl 1210.65185 Cai, Zhiqiang; Zhang, Shun 35 2010 First-order system least squares for the stress-displacement formulation: Linear elasticity. Zbl 1063.74099 Cai, Zhiqiang; Starke, Gerhard 33 2003 Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs. Zbl 07506625 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min; Liu, Xinyu 31 2020 Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance. Zbl 1366.93565 Cai, Z.; de Queiroz, M. S.; Dawson, D. M. 29 2006 Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimates. Zbl 1359.65248 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 29 2017 First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity. Zbl 0856.76010 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F. 28 1995 Robust equilibrated residual error estimator for diffusion problems: conforming elements. Zbl 1253.65175 Cai, Zhiqiang; Zhang, Shun 26 2012 A sufficiently smooth projection operator. Zbl 1366.93283 Cai, Z.; de Queiroz, M. S.; Dawson, D. M. 26 2006 Nonconforming quadrilateral finite elements: a correction. Zbl 1012.65124 Cai, Z.; Douglas, J. jun.; Santos, J. E.; Sheen, D.; Ye, X. 25 2000 First-order system \(\mathcal L\mathcal L^*\) (FOSLL*): Scalar elliptic partial differential equations. Zbl 1008.65085 Cai, Z.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. 23 2001 A multigrid method for the pseudostress formulation of Stokes problems. Zbl 1182.76896 Cai, Zhiqiang; Wang, Yanqiu 22 2007 First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. Zbl 0968.74061 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F.; Parter, Seymour V. 21 1998 An adaptive least squares mixed finite element method for the stress-displacement formulation of linear elasticity. Zbl 1061.74050 Cai, Zhiqiang; Korsawe, Johannes; Starke, Gerhard 21 2005 Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. Zbl 1410.76160 Cai, Zhiqiang; Wang, Chunbo; Zhang, Shun 20 2010 Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations. Zbl 1230.65117 Cai, Zhiqiang; Zhang, Shun 20 2010 An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type. Zbl 1274.76246 Cai, Z.; Westphal, C. R. 20 2009 The \(L^2\) norm error estimates for the div least-squares method. Zbl 1138.76053 Cai, Zhiqiang; Ku, Jaeun 18 2006 Multigrid methods for nearly singular linear equations and eigenvalue problems. Zbl 0873.65030 Cai, Zhiqiang; Mandel, Jan; McCormick, Steve 17 1997 Solution methods for the Poisson equation with corner singularities: Numerical results. Zbl 0991.65117 Cai, Zhiqiang; Kim, Seokchan; Shin, Byeong-Chun 16 2001 Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation. Zbl 1307.76050 Cai, Zhiqiang; Zhang, Shun 16 2012 A mixed nonconforming finite element for linear elasticity. Zbl 1076.74053 Cai, Zhiqiang; Ye, Xiu 15 2005 Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach. Zbl 1416.65268 Hao, Zhaopeng; Park, Moongyu; Lin, Guang; Cai, Zhiqiang 15 2019 A finite element method using singular functions for Poisson equations: Mixed boundary conditions. Zbl 1124.65108 Cai, Zhiqiang; Kim, Seokchan; Kim, Sangdong; Kong, Sooryun 14 2006 Multilevel iteration for mixed finite element systems with penalty. Zbl 0809.65114 Cai, Zhiqiang; Goldstein, Charles I.; Pasciak, Joseph E. 14 1993 A recovery-based a posteriori error estimator for \(H\)(curl) interface problems. Zbl 1425.65153 Cai, Zhiqiang; Cao, Shuhao 12 2015 Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Zbl 1352.76043 Cai, Zhiqiang; Wang, Yanqiu 11 2010 A weighted \(H\)(div) least-squares method for second-order elliptic problems. Zbl 1168.65069 Cai, Z.; Westphal, C. R. 11 2008 Recovery-based error estimator for stabilized finite element methods for the Stokes equation. Zbl 1296.76087 Song, Lina; Hou, Yanren; Cai, Zhiqiang 11 2014 Analysis of epidemic vaccination strategies on heterogeneous networks: based on SEIRV model and evolutionary game. Zbl 1510.92223 Meng, Xueyu; Cai, Zhiqiang; Si, Shubin; Duan, Dongli 11 2021 Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation. Zbl 07515414 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 11 2021 Residual-based a posteriori error estimate for interface problems: nonconforming linear elements. Zbl 1355.65145 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 10 2017 Div first-order system LL* (FOSLL*) for second-order elliptic partial differential equations. Zbl 1312.65005 Cai, Zhiqiang; Falgout, Rob; Zhang, Shun 10 2015 First-order system least squares for linear elasticity: Numerical results. Zbl 0988.74061 Cai, Z.; Lee, C.-O.; Manteuffel, T. A.; McCormick, S. F. 9 2000 Improved ZZ a posteriori error estimators for diffusion problems: conforming linear elements. Zbl 1439.65153 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 9 2017 Optimal error estimate for the div least-squares method with data \(f\in L^2\) and application to nonlinear problems. Zbl 1275.65079 Cai, Zhiqiang; Ku, Jaeun 8 2010 First-order system least squares for the Stokes and linear elasticity equations: Further results. Zbl 0968.76035 Cai, Z.; Lee, C.-O.; Manteuffel, T. A.; McCormick, S. F. 8 2000 Least-squares finite element approximations for the Reissner-Mindlin plate. Zbl 0982.74062 Chai, Zhiqiang; Ye, Xiu; Zhang, Huilong 8 1999 Adaptive two-layer ReLU neural network. I: Best least-squares approximation. Zbl 1504.65275 Liu, Min; Cai, Zhiqiang; Chen, Jingshuang 8 2022 An error estimate for two-dimensional Stokes driven cavity flow. Zbl 1198.65217 Cai, Zhiqiang; Wang, Yanqiu 7 2009 Extended Bose-Hubbard models with ultracold magnetic atoms. Zbl 1355.81173 Baier, S.; Mark, M. J.; Petter, D.; Aikawa, K.; Chomaz, L.; Cai, Z.; Baranov, M.; Zoller, P.; Ferlaino, F. 6 2016 A dual finite element method for a singularly perturbed reaction-diffusion problem. Zbl 1440.65188 Cai, Zhiqiang; Ku, JaEun 6 2020 Least squares for the perturbed Stokes equations and the Reissner–Mindlin plate. Zbl 1003.76047 Cai, Zhiqiang 5 2000 A least-squares finite element approximation for the compressible Stokes equations. Zbl 0965.76039 Cai, Zhiqiang; Ye, Xiu 5 2000 Domain decomposition for a mixed finite element method in three dimensions. Zbl 1057.65095 Cai, Z.; Parashkevov, R. R.; Russell, T. F.; Wilson, J. D.; Ye, X. 5 2003 Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems. Zbl 1447.65130 Cai, Difeng; Cai, Zhiqiang; Zhang, Shun 5 2020 Adaptive two-layer ReLU neural network. II: Ritz approximation to elliptic PDEs. Zbl 1524.65939 Liu, Min; Cai, Zhiqiang 5 2022 Goal-oriented local a posteriori error estimators for \(H(\text{div})\) least-squares finite element methods. Zbl 1298.65147 Cai, Zhiqiang; Ku, JaEun 4 2011 Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps. Zbl 1245.65148 Cai, Zhiqiang; Zhang, Shun 4 2012 Least-squares method for the Oseen equation. Zbl 1407.76026 Cai, Zhiqiang; Chen, Binghe 4 2016 Least-squares methods for elasticity and Stokes equations with weakly imposed symmetry. Zbl 1420.65115 Bertrand, Fleurianne; Cai, Zhiqiang; Park, Eun Young 4 2019 Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global \(L_2\) norm. Zbl 1443.65315 Cai, Zhiqiang; Carey, Varis; Ku, JaEun; Park, Eun-Jae 4 2015 First-order system least squares for the Signorini contact problem in linear elasticity. Zbl 1410.74058 Attia, Frank S.; Cai, Zhiqiang; Starke, Gerhard 4 2009 A hybrid a posteriori error estimator for conforming finite element approximations. Zbl 1440.65187 Cai, Difeng; Cai, Zhiqiang 4 2018 Robust equilibrated error estimator for diffusion problems: mixed finite elements in two dimensions. Zbl 1437.65177 Cai, Difeng; Cai, Zhiqiang; Zhang, Shun 4 2020 Generalized Prager-Synge identity and robust equilibrated error estimators for discontinuous elements. Zbl 1478.65116 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 4 2021 A finite element method using singular functions: interface problems. Zbl 1140.65084 Kim, Seokchan; Cai, Zhiqiang; Pyo, Jae-Hong; Kong, Sooryoun 3 2007 Improved ZZ a posteriori error estimators for diffusion problems: discontinuous elements. Zbl 1460.65143 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 3 2021 Robust a posteriori error estimation for finite element approximation to \(\boldsymbol{H}(\mathbf{curl})\) problem. Zbl 1439.78019 Cai, Zhiqiang; Cao, Shuhao; Falgout, Rob 3 2016 Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law. Zbl 1484.65174 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 3 2022 The discrete first-order system least squares: The second-order elliptic boundary value problem. Zbl 1017.65092 Cai, Zhiqiang; Shin, Byeong Chun 2 2002 Analytic spectra of CMB anisotropies and polarization generated by scalar perturbations in synchronous gauge. Zbl 1242.85011 Cai, Z.; Zhang, Y. 2 2012 Self-adaptive deep neural network: numerical approximation to functions and PDEs. Zbl 07518092 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 2 2022 Error estimate of a finite element method using stress intensity factor. Zbl 1442.65348 Cai, Zhiqiang; Kim, Seokchan; Lee, Hyung-Chun 2 2018 Error estimates for a Schwarz alternating procedure on L-shaped regions. Zbl 0655.65113 Cai, Zhiqiang; Guo, Benju 1 1988 Uniform asymptotic analysis for transient waves in a pre-stressed compressible hyperelastic rod. Zbl 0982.74038 Dai, H.-H.; Cai, Z. 1 2000 An analytic basis for multigrid methods for stabilized finite element methods for the Stokes problem. Zbl 0911.65124 Cai, Z.; Douglas, J. jun. 1 1997 A finite element method using singular functions for the Poisson equation: crack singularities. Zbl 1071.65152 Cai, Zhiqiang; Kim, Seokchan; Woo, Gyungsoo 1 2002 Convergence estimates of multilevel additive and multiplicative algorithms for non-symmetric and indefinite problems. Zbl 0852.65112 Cai, Zhiqiang; Lai, Chen-Yao G. 1 1996 Stabilized finite element methods with fast iterative solution algorithms for the Stokes problem. Zbl 0943.76048 Cai, Zhiqiang; Douglas, Jim jun. 1 1998 Computational complexity of the Schwarz alternating procedure. Zbl 0726.65029 Cai, Zhiqiang; McCormick, Steve 1 1989 Norm estimates of product operators with application to domain decomposition. Zbl 0766.65100 Cai, Zhiqiang 1 1993 Joint integrated importance measure for multi-state transition systems. Zbl 1284.90018 Si, Shubin; Dui, Hongyan; Cai, Zhiqiang; Sun, Shudong; Zhang, Yingfeng 1 2012 A hidden semi-Markov model with duration-dependent state transition probabilities for prognostics. Zbl 1407.90336 Wang, Ning; Sun, Shu-dong; Cai, Zhi-qiang; Zhang, Shuai; Saygin, Can 1 2014 A finite element method for Dirichlet boundary control of elliptic partial differential equations. Zbl 1486.49032 Du, Shaohong; Cai, Zhiqiang 1 2022 Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: discrete divergence operator. Zbl 07715661 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 1 2023 Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: discrete divergence operator. Zbl 07715661 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 1 2023 Adaptive two-layer ReLU neural network. I: Best least-squares approximation. Zbl 1504.65275 Liu, Min; Cai, Zhiqiang; Chen, Jingshuang 8 2022 Adaptive two-layer ReLU neural network. II: Ritz approximation to elliptic PDEs. Zbl 1524.65939 Liu, Min; Cai, Zhiqiang 5 2022 Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law. Zbl 1484.65174 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 3 2022 Self-adaptive deep neural network: numerical approximation to functions and PDEs. Zbl 07518092 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 2 2022 A finite element method for Dirichlet boundary control of elliptic partial differential equations. Zbl 1486.49032 Du, Shaohong; Cai, Zhiqiang 1 2022 Analysis of epidemic vaccination strategies on heterogeneous networks: based on SEIRV model and evolutionary game. Zbl 1510.92223 Meng, Xueyu; Cai, Zhiqiang; Si, Shubin; Duan, Dongli 11 2021 Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation. Zbl 07515414 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min 11 2021 Generalized Prager-Synge identity and robust equilibrated error estimators for discontinuous elements. Zbl 1478.65116 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 4 2021 Improved ZZ a posteriori error estimators for diffusion problems: discontinuous elements. Zbl 1460.65143 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 3 2021 Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs. Zbl 07506625 Cai, Zhiqiang; Chen, Jingshuang; Liu, Min; Liu, Xinyu 31 2020 A dual finite element method for a singularly perturbed reaction-diffusion problem. Zbl 1440.65188 Cai, Zhiqiang; Ku, JaEun 6 2020 Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems. Zbl 1447.65130 Cai, Difeng; Cai, Zhiqiang; Zhang, Shun 5 2020 Robust equilibrated error estimator for diffusion problems: mixed finite elements in two dimensions. Zbl 1437.65177 Cai, Difeng; Cai, Zhiqiang; Zhang, Shun 4 2020 Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach. Zbl 1416.65268 Hao, Zhaopeng; Park, Moongyu; Lin, Guang; Cai, Zhiqiang 15 2019 Least-squares methods for elasticity and Stokes equations with weakly imposed symmetry. Zbl 1420.65115 Bertrand, Fleurianne; Cai, Zhiqiang; Park, Eun Young 4 2019 A hybrid a posteriori error estimator for conforming finite element approximations. Zbl 1440.65187 Cai, Difeng; Cai, Zhiqiang 4 2018 Error estimate of a finite element method using stress intensity factor. Zbl 1442.65348 Cai, Zhiqiang; Kim, Seokchan; Lee, Hyung-Chun 2 2018 Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimates. Zbl 1359.65248 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 29 2017 Residual-based a posteriori error estimate for interface problems: nonconforming linear elements. Zbl 1355.65145 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 10 2017 Improved ZZ a posteriori error estimators for diffusion problems: conforming linear elements. Zbl 1439.65153 Cai, Zhiqiang; He, Cuiyu; Zhang, Shun 9 2017 Extended Bose-Hubbard models with ultracold magnetic atoms. Zbl 1355.81173 Baier, S.; Mark, M. J.; Petter, D.; Aikawa, K.; Chomaz, L.; Cai, Z.; Baranov, M.; Zoller, P.; Ferlaino, F. 6 2016 Least-squares method for the Oseen equation. Zbl 1407.76026 Cai, Zhiqiang; Chen, Binghe 4 2016 Robust a posteriori error estimation for finite element approximation to \(\boldsymbol{H}(\mathbf{curl})\) problem. Zbl 1439.78019 Cai, Zhiqiang; Cao, Shuhao; Falgout, Rob 3 2016 A recovery-based a posteriori error estimator for \(H\)(curl) interface problems. Zbl 1425.65153 Cai, Zhiqiang; Cao, Shuhao 12 2015 Div first-order system LL* (FOSLL*) for second-order elliptic partial differential equations. Zbl 1312.65005 Cai, Zhiqiang; Falgout, Rob; Zhang, Shun 10 2015 Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global \(L_2\) norm. Zbl 1443.65315 Cai, Zhiqiang; Carey, Varis; Ku, JaEun; Park, Eun-Jae 4 2015 Recovery-based error estimator for stabilized finite element methods for the Stokes equation. Zbl 1296.76087 Song, Lina; Hou, Yanren; Cai, Zhiqiang 11 2014 A hidden semi-Markov model with duration-dependent state transition probabilities for prognostics. Zbl 1407.90336 Wang, Ning; Sun, Shu-dong; Cai, Zhi-qiang; Zhang, Shuai; Saygin, Can 1 2014 Robust equilibrated residual error estimator for diffusion problems: conforming elements. Zbl 1253.65175 Cai, Zhiqiang; Zhang, Shun 26 2012 Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation. Zbl 1307.76050 Cai, Zhiqiang; Zhang, Shun 16 2012 Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps. Zbl 1245.65148 Cai, Zhiqiang; Zhang, Shun 4 2012 Analytic spectra of CMB anisotropies and polarization generated by scalar perturbations in synchronous gauge. Zbl 1242.85011 Cai, Z.; Zhang, Y. 2 2012 Joint integrated importance measure for multi-state transition systems. Zbl 1284.90018 Si, Shubin; Dui, Hongyan; Cai, Zhiqiang; Sun, Shudong; Zhang, Yingfeng 1 2012 Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. Zbl 1232.65142 Cai, Zhiqiang; Ye, Xiu; Zhang, Shun 79 2011 Goal-oriented local a posteriori error estimators for \(H(\text{div})\) least-squares finite element methods. Zbl 1298.65147 Cai, Zhiqiang; Ku, JaEun 4 2011 Mixed finite element methods for incompressible flow: stationary Stokes equations. Zbl 1267.76059 Cai, Zhiqiang; Tong, Charles; Vassilevski, Panayot S.; Wang, Chunbo 43 2010 Recovery-based error estimators for interface problems: mixed and nonconforming finite elements. Zbl 1210.65185 Cai, Zhiqiang; Zhang, Shun 35 2010 Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. Zbl 1410.76160 Cai, Zhiqiang; Wang, Chunbo; Zhang, Shun 20 2010 Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations. Zbl 1230.65117 Cai, Zhiqiang; Zhang, Shun 20 2010 Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Zbl 1352.76043 Cai, Zhiqiang; Wang, Yanqiu 11 2010 Optimal error estimate for the div least-squares method with data \(f\in L^2\) and application to nonlinear problems. Zbl 1275.65079 Cai, Zhiqiang; Ku, Jaeun 8 2010 Recovery-based error estimator for interface problems: Conforming linear elements. Zbl 1204.65129 Cai, Zhiqiang; Zhang, Shun 53 2009 An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type. Zbl 1274.76246 Cai, Z.; Westphal, C. R. 20 2009 An error estimate for two-dimensional Stokes driven cavity flow. Zbl 1198.65217 Cai, Zhiqiang; Wang, Yanqiu 7 2009 First-order system least squares for the Signorini contact problem in linear elasticity. Zbl 1410.74058 Attia, Frank S.; Cai, Zhiqiang; Starke, Gerhard 4 2009 A weighted \(H\)(div) least-squares method for second-order elliptic problems. Zbl 1168.65069 Cai, Z.; Westphal, C. R. 11 2008 A multigrid method for the pseudostress formulation of Stokes problems. Zbl 1182.76896 Cai, Zhiqiang; Wang, Yanqiu 22 2007 A finite element method using singular functions: interface problems. Zbl 1140.65084 Kim, Seokchan; Cai, Zhiqiang; Pyo, Jae-Hong; Kong, Sooryoun 3 2007 Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance. Zbl 1366.93565 Cai, Z.; de Queiroz, M. S.; Dawson, D. M. 29 2006 A sufficiently smooth projection operator. Zbl 1366.93283 Cai, Z.; de Queiroz, M. S.; Dawson, D. M. 26 2006 The \(L^2\) norm error estimates for the div least-squares method. Zbl 1138.76053 Cai, Zhiqiang; Ku, Jaeun 18 2006 A finite element method using singular functions for Poisson equations: Mixed boundary conditions. Zbl 1124.65108 Cai, Zhiqiang; Kim, Seokchan; Kim, Sangdong; Kong, Sooryun 14 2006 An adaptive least squares mixed finite element method for the stress-displacement formulation of linear elasticity. Zbl 1061.74050 Cai, Zhiqiang; Korsawe, Johannes; Starke, Gerhard 21 2005 A mixed nonconforming finite element for linear elasticity. Zbl 1076.74053 Cai, Zhiqiang; Ye, Xiu 15 2005 Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. Zbl 1159.76347 Cai, Zhiqiang; Lee, Barry; Wang, Ping 57 2004 Least-squares methods for linear elasticity. Zbl 1159.74419 Cai, Zhiqiang; Starke, Gerhard 56 2004 Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Zbl 1020.65087 Cai, Zhiqiang; Douglas, Jim jun.; Park, Moongyu 40 2003 First-order system least squares for the stress-displacement formulation: Linear elasticity. Zbl 1063.74099 Cai, Zhiqiang; Starke, Gerhard 33 2003 Domain decomposition for a mixed finite element method in three dimensions. Zbl 1057.65095 Cai, Z.; Parashkevov, R. R.; Russell, T. F.; Wilson, J. D.; Ye, X. 5 2003 The discrete first-order system least squares: The second-order elliptic boundary value problem. Zbl 1017.65092 Cai, Zhiqiang; Shin, Byeong Chun 2 2002 A finite element method using singular functions for the Poisson equation: crack singularities. Zbl 1071.65152 Cai, Zhiqiang; Kim, Seokchan; Woo, Gyungsoo 1 2002 A finite element method using singular functions for the Poisson equation: Corner singularities. Zbl 0992.65122 Cai, Zhiqiang; Kim, Seokchan 43 2001 First-order system \(\mathcal L\mathcal L^*\) (FOSLL*): Scalar elliptic partial differential equations. Zbl 1008.65085 Cai, Z.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. 23 2001 Solution methods for the Poisson equation with corner singularities: Numerical results. Zbl 0991.65117 Cai, Zhiqiang; Kim, Seokchan; Shin, Byeong-Chun 16 2001 Nonconforming quadrilateral finite elements: a correction. Zbl 1012.65124 Cai, Z.; Douglas, J. jun.; Santos, J. E.; Sheen, D.; Ye, X. 25 2000 First-order system least squares for linear elasticity: Numerical results. Zbl 0988.74061 Cai, Z.; Lee, C.-O.; Manteuffel, T. A.; McCormick, S. F. 9 2000 First-order system least squares for the Stokes and linear elasticity equations: Further results. Zbl 0968.76035 Cai, Z.; Lee, C.-O.; Manteuffel, T. A.; McCormick, S. F. 8 2000 Least squares for the perturbed Stokes equations and the Reissner–Mindlin plate. Zbl 1003.76047 Cai, Zhiqiang 5 2000 A least-squares finite element approximation for the compressible Stokes equations. Zbl 0965.76039 Cai, Zhiqiang; Ye, Xiu 5 2000 Uniform asymptotic analysis for transient waves in a pre-stressed compressible hyperelastic rod. Zbl 0982.74038 Dai, H.-H.; Cai, Z. 1 2000 Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Zbl 0922.65071 Hou, Thomas Y.; Wu, Xiao-Hui; Cai, Zhiqiang 215 1999 A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Zbl 0947.76047 Cai, Zhiqiang; Douglas, Jim jun.; Ye, Xiu 57 1999 Least-squares finite element approximations for the Reissner-Mindlin plate. Zbl 0982.74062 Chai, Zhiqiang; Ye, Xiu; Zhang, Huilong 8 1999 Analysis of velocity-flux first-order system least-squares principles for the Navier-Stokes equations. I. Zbl 0910.76032 Bochev, P.; Cai, Z.; Manteuffel, T. A.; McCormick, S. F. 48 1998 First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. Zbl 0968.74061 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F.; Parter, Seymour V. 21 1998 Stabilized finite element methods with fast iterative solution algorithms for the Stokes problem. Zbl 0943.76048 Cai, Zhiqiang; Douglas, Jim jun. 1 1998 First-order system least squares for second-order partial differential equations: Part II. Zbl 0912.65089 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F. 86 1997 First-order system least squares for the Stokes equations, with application to linear elasticity. Zbl 0901.76052 Cai, Z.; Manteuffel, T. A.; McCormick, S. F. 76 1997 Control-volume mixed finite element methods. Zbl 0941.76050 Cai, Z.; Jones, J. E.; McCormick, S. F.; Russell, T. F. 50 1997 Multigrid methods for nearly singular linear equations and eigenvalue problems. Zbl 0873.65030 Cai, Zhiqiang; Mandel, Jan; McCormick, Steve 17 1997 An analytic basis for multigrid methods for stabilized finite element methods for the Stokes problem. Zbl 0911.65124 Cai, Z.; Douglas, J. jun. 1 1997 Convergence estimates of multilevel additive and multiplicative algorithms for non-symmetric and indefinite problems. Zbl 0852.65112 Cai, Zhiqiang; Lai, Chen-Yao G. 1 1996 First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity. Zbl 0856.76010 Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F. 28 1995 First-order system least squares for second-order partial differential equations. I. Zbl 0813.65119 Cai, Z.; Lazarov, R.; Manteuffel, T. A.; McCormick, S. F. 144 1994 Multilevel iteration for mixed finite element systems with penalty. Zbl 0809.65114 Cai, Zhiqiang; Goldstein, Charles I.; Pasciak, Joseph E. 14 1993 Norm estimates of product operators with application to domain decomposition. Zbl 0766.65100 Cai, Zhiqiang 1 1993 On the finite volume element method. Zbl 0731.65093 Cai, Zhiqiang 233 1991 The finite volume element method for diffusion equations on general triangulations. Zbl 0729.65086 Cai, Zhiqiang; Mandel, Jan; McCormick, Steve 123 1991 On the accuracy of the finite volume element method for diffusion equations on composite grids. Zbl 0707.65073 Cai, Zhiqiang; McCormick, Steve 91 1990 Computational complexity of the Schwarz alternating procedure. Zbl 0726.65029 Cai, Zhiqiang; McCormick, Steve 1 1989 Error estimates for a Schwarz alternating procedure on L-shaped regions. Zbl 0655.65113 Cai, Zhiqiang; Guo, Benju 1 1988 all cited Publications top 5 cited Publications all top 5 Cited by 1,635 Authors 41 Cai, Zhiqiang 22 Zou, Qingsong 21 Zhang, Zhimin 20 Li, Yonghai 20 Zhang, Shun 19 Yang, Suh-Yuh 18 Chen, Chuanjun 18 Rui, Hongxing 18 Yuan, Yirang 17 Cui, Junzhi 16 Bi, Chunjia 16 Luo, Zhendong 16 Mu, Lin 16 Ye, Xiu 15 Chen, Zhangxin 14 Chen, Yanping 14 Li, Jian 14 Starke, Gerhard 13 Bertrand, Fleurianne 13 Li, Changfeng 13 Shi, Dongyang 12 Gatica, Gabriel N. 12 He, Cuiyu 12 Kim, Seokchan 12 Ku, Jaeun 12 Lee, Eunjung 12 Manteuffel, Thomas A. 12 Park, Eun-Jae 12 Schröder, Jörg 12 Yang, Min 11 Carstensen, Carsten 11 Feng, Xinlong 11 Huang, Yunqing 11 Sheen, Dongwoo 11 Vassilevski, Panayot Spirov 10 Abdulle, Assyr 10 Cao, Waixiang 10 Guo, Hui 10 He, Wenming 10 Hou, Thomas Yizhao 10 Kim, Sangdong 10 Li, Hong 10 Ruiz-Baier, Ricardo 10 Schwarz, Alexander 10 Wang, Junping 10 Wu, Jiming 10 Zhang, Zhiyue 9 Chen, Jinru 9 Lazarov, Raytcho D. 9 Valentin, Frédéric 9 Xu, Jinchao 9 Zhou, Yanhui 8 Cao, Liqun 8 Demkowicz, Leszek F. 8 Deng, Weibing 8 Guan, Xiaofei 8 Hu, Jun 8 Lin, Yanping 8 Liu, Min 8 Liu, Wei 8 McCormick, Steve Fahrney 8 Oyarzúa, Ricardo 8 Song, Huailing 8 Zhang, Tie 8 Zhang, Zhiwen 7 Cangiani, Andrea 7 Cao, Shuhao 7 Dehghan Takht Fooladi, Mehdi 7 Lee, Hsueh-Chen 7 Mao, Shipeng 7 Shin, Byeong Chun 7 Wang, Xiang 7 Xie, Xiaoping 7 Yang, Qing 7 Zhang, Xu 6 Adler, James H. 6 Bringmann, Philipp 6 Chen, Jingshuang 6 Chen, Shaochun 6 Choi, Hyungjun 6 Dong, Hao 6 Duan, Huoyuan 6 Führer, Thomas 6 Gerritsma, Marc I. 6 Ginting, Victor 6 He, Yinnian 6 Hessari, Peyman 6 Huang, Pengzhan 6 Kumar, Sarvesh 6 Kwak, Do Young 6 Li, Minghao 6 Li, Qin 6 Luo, Zhongxuan 6 Lv, Junliang 6 Meng, Zhaoliang 6 Mora, David 6 Saad, Mazen 6 Sinha, Rajen Kumar 6 Song, Lina 6 Wang, Hong ...and 1,535 more Authors all top 5 Cited in 152 Serials 107 Journal of Computational and Applied Mathematics 101 Computers & Mathematics with Applications 100 Computer Methods in Applied Mechanics and Engineering 80 Journal of Computational Physics 72 Journal of Scientific Computing 67 Applied Numerical Mathematics 65 Numerical Methods for Partial Differential Equations 62 Applied Mathematics and Computation 44 Numerische Mathematik 41 SIAM Journal on Numerical Analysis 35 Mathematics of Computation 22 SIAM Journal on Scientific Computing 22 Advances in Computational Mathematics 22 Multiscale Modeling & Simulation 19 International Journal for Numerical Methods in Engineering 19 International Journal of Computer Mathematics 19 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 18 M\(^3\)AS. Mathematical Models & Methods in Applied Sciences 15 Advances in Applied Mathematics and Mechanics 13 Mathematical Methods in the Applied Sciences 13 Computational Methods in Applied Mathematics 11 Journal of Mathematical Analysis and Applications 11 Numerical Algorithms 11 Numerical Linear Algebra with Applications 11 Computational Geosciences 10 Numerical Functional Analysis and Optimization 10 Journal of Numerical Mathematics 10 International Journal of Numerical Analysis and Modeling 10 Communications in Computational Physics 9 Computing 9 Science China. Mathematics 8 Calcolo 7 Computing and Visualization in Science 7 M2AN. Mathematical Modelling and Numerical Analysis. 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Series B 3 Communications in Numerical Methods in Engineering 3 Computational and Applied Mathematics 3 ETNA. Electronic Transactions on Numerical Analysis 3 Engineering Analysis with Boundary Elements 3 Journal of Inequalities and Applications 3 Discrete and Continuous Dynamical Systems. Series B 3 Acta Mathematica Scientia. Series B. (English Edition) 3 Boletim da Sociedade Paranaense de Matemática. Terceira Série 3 Advances in Difference Equations 3 East Asian Mathematical Journal 3 Journal of the Korean Society for Industrial and Applied Mathematics 2 Acta Mechanica 2 Journal of Fluid Mechanics 2 Applied Mathematics Letters 2 Mathematical and Computer Modelling 2 Science in China. Series A 2 Japan Journal of Industrial and Applied Mathematics 2 Journal of Elasticity 2 Applied and Computational Harmonic Analysis 2 International Journal of Computational Fluid Dynamics 2 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 2 Honam Mathematical Journal 2 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 2 Communications in Nonlinear Science and Numerical Simulation 2 Journal of Applied Mathematics 2 Communications on Pure and Applied Analysis 2 Central European Journal of Mathematics 2 Acta Numerica 2 GAMM-Mitteilungen 2 Communications in Applied Mathematics and Computational Science 2 Computational & Mathematical Methods in Medicine 2 Discrete and Continuous Dynamical Systems. Series S 2 Journal of Mathematics 2 AMM. Applied Mathematics and Mechanics. (English Edition) 2 AIMS Mathematics 2 SMAI Journal of Computational Mathematics 2 Communications on Applied Mathematics and Computation 2 International Journal on Finite Volumes 1 Computers and Fluids 1 Computer Physics Communications 1 International Journal for Numerical and Analytical Methods in Geomechanics 1 Indian Journal of Pure & Applied Mathematics 1 Journal of Engineering Mathematics 1 Journal of the Franklin Institute 1 Journal of Mathematical Physics ...and 52 more Serials all top 5 Cited in 39 Fields 1,179 Numerical analysis (65-XX) 676 Partial differential equations (35-XX) 397 Fluid mechanics (76-XX) 184 Mechanics of deformable solids (74-XX) 37 Calculus of variations and optimal control; optimization (49-XX) 32 Computer science (68-XX) 32 Biology and other natural sciences (92-XX) 24 Classical thermodynamics, heat transfer (80-XX) 23 Optics, electromagnetic theory (78-XX) 19 Statistical mechanics, structure of matter (82-XX) 16 Ordinary differential equations (34-XX) 15 Approximations and expansions (41-XX) 12 Probability theory and stochastic processes (60-XX) 8 Real functions (26-XX) 6 Potential theory (31-XX) 5 Integral equations (45-XX) 5 Global analysis, analysis on manifolds (58-XX) 5 Statistics (62-XX) 5 Geophysics (86-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 4 Special functions (33-XX) 4 Operator theory (47-XX) 4 Quantum theory (81-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Harmonic analysis on Euclidean spaces (42-XX) 3 Functional analysis (46-XX) 3 Operations research, mathematical programming (90-XX) 3 Systems theory; control (93-XX) 2 History and biography (01-XX) 2 Combinatorics (05-XX) 2 Number theory (11-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Difference and functional equations (39-XX) 2 Mechanics of particles and systems (70-XX) 2 Information and communication theory, circuits (94-XX) 1 Measure and integration (28-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 Astronomy and astrophysics (85-XX) Citations by Year