American Institute of Mathematical Sciences

Advanced
October  2010, 26(4): 1305-1318. doi: 10.3934/dcds.2010.26.1305

Maximum norm error estimates for Div least-squares method for Darcy flows

JaEun Ku 1,

1. 

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States

Received  November 2008 Revised  April 2009 Published  December 2009

Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].
Keywords: elliptic equations., least-squares methods, Galerkin methods, maximum norm error estimate.
Mathematics Subject Classification: Primary: 65M60, 65M1.
Citation: JaEun Ku. Maximum norm error estimates for Div least-squares method for Darcy flows. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1305-1318. doi: 10.3934/dcds.2010.26.1305
[1]

Hassan Mohammad, Mohammed Yusuf Waziri, Sandra Augusta Santos. A brief survey of methods for solving nonlinear least-squares problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 1-13. doi: 10.3934/naco.2019001
[2]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15
[3]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[4]

Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows. Electronic Research Archive, 2021, 29 (4) : 2755-2770. doi: 10.3934/era.2021012
[5]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
[6]

Mila Nikolova. Analytical bounds on the minimizers of (nonconvex) regularized least-squares. Inverse Problems & Imaging, 2008, 2 (1) : 133-149. doi: 10.3934/ipi.2008.2.133
[7]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340
[8]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[9]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325
[10]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[11]

Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189
[12]

Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350
[13]

H. D. Scolnik, N. E. Echebest, M. T. Guardarucci. Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 175-191. doi: 10.3934/jimo.2009.5.175
[14]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[15]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126
[16]

Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043
[17]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[18]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[19]

Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic & Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373
[20]

Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences