May  2009, 11(3): 669-690. doi: 10.3934/dcdsb.2009.11.669

Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization

1. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States

2. 

Department of Statistics, University of South Carolina, Columbia, SC 29208, United States

3. 

School of Mathematics and System Sciences, Shandong University, Jinan, Shandong 250100, China

Received  February 2008 Revised  May 2008 Published  March 2009

An adaptive algorithm for steady convection-diffusion problems that combines a posteriori error estimation with conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) is proposed and tested in two dimensional domains. Different from most current adaptive methods, this algorithm realizes mesh refinement and coarsening implicitly at each level by nodes insertion and redistribution. Especially, the nodes redistribution is implemented through the generation of CfCVDTs with some density function derived from the a posteriori error estimators for the problem. Numerical experiments show that the convergence rates achieved are almost the best obtainable using the linear finite volume discretizations and the resulting meshes always maintain high quality.
Citation: Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669
[1]

Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309

[2]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

[3]

Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471

[4]

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

[5]

Qiang Du, Zhan Huang, Richard B. Lehoucq. Nonlocal convection-diffusion volume-constrained problems and jump processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 373-389. doi: 10.3934/dcdsb.2014.19.373

[6]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[7]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11

[8]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004

[9]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689

[10]

Iryna Pankratova, Andrey Piatnitski. Homogenization of convection-diffusion equation in infinite cylinder. Networks & Heterogeneous Media, 2011, 6 (1) : 111-126. doi: 10.3934/nhm.2011.6.111

[11]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873

[12]

Phillip Colella. High-order finite-volume methods on locally-structured grids. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4247-4270. doi: 10.3934/dcds.2016.36.4247

[13]

Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935

[14]

M. González, J. Jansson, S. Korotov. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Conference Publications, 2015, 2015 (special) : 525-532. doi: 10.3934/proc.2015.0525

[15]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133

[16]

Giacomo Frassoldati, Luca Zanni, Gaetano Zanghirati. New adaptive stepsize selections in gradient methods. Journal of Industrial & Management Optimization, 2008, 4 (2) : 299-312. doi: 10.3934/jimo.2008.4.299

[17]

Cesare Bracco, Annalisa Buffa, Carlotta Giannelli, Rafael Vázquez. Adaptive isogeometric methods with hierarchical splines: An overview. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 241-261. doi: 10.3934/dcds.2019010

[18]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

[19]

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

[20]

Christopher Rackauckas, Qing Nie. Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2731-2761. doi: 10.3934/dcdsb.2017133

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]