Abstract
Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analysis.
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Xu, J. C. and Zhou, A. H. Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69(231), 881–909 (1999)
Xu, J. C. and Zhou, A. H. Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14(4), 293–327 (2001)
Xu, J. C. and Zhou, A. H. Some local and parallel properties of finite element discretizations. Proceedings for Eleventh International Conference on Domain Decomposition Methods (eds. Lai, C. H., Bjφsted, P. E., Cross, M., and Widlund, O. B.), Greenwich, England, 140–147 (1999)
He, Y. N., Xu, J. C., and Zhou, A. H. Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109(3), 415–434 (2008)
He, Y. N., Xu, J. C., and Zhou, A. H. Local and parallel finite element algorithms for the Navier-Stokes problem. J. Comput. Math. 24(3), 227–238 (2006)
Ma, F. Y., Ma, Y. C., and Wo, W. F. Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed. 28(1), 27–35 (2007) DOI: 10.1007/s10483-007-0104-x
Xu, J. C. A novel two-grid method for semilinear equations. SIAM J. Sci. Comput. 15(1), 231–237 (1994)
Xu, J. C. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)
Heywood, J. G. and Rannacher, R. Finite element approximation of the nonstationary Navier-Stokes problem, part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353–384 (1990)
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(Communicated by Zhe-wei ZHOU)
Project supported by the National Natural Science Foundation of China (No. 10871156) and the Program for New Century Excellent Talents in University (No. NCET-06-0829)
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Liu, Qf., Hou, Yr. Local and parallel finite element algorithms for time-dependent convection-diffusion equations. Appl. Math. Mech.-Engl. Ed. 30, 787–794 (2009). https://doi.org/10.1007/s10483-009-0613-x
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DOI: https://doi.org/10.1007/s10483-009-0613-x