一种求解半线性椭圆问题的快速多重网格法
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摘要
本文介绍一种求解半线性问题的完全多重网格算法,该算法是基于多重校正算法与线性边值问题的多重网格迭代结合而设计的.多重校正算法将半线性问题的求解转化成线性边值问题的求解加上在一个低维空间上的半线性问题的求解.利用并行计算技术,这里所提出的多重网格算法可以明显地提高求解半线性椭圆问题的效率.更进一步,当非线性项是多项式函数的时候,本文也设计了一种高效的完全多重网格算法,并且通过分析可以知道该算法求解多项式形式的半线性椭圆问题的计算量具有渐近最优的性质.最后用数值实验验证了本文算法的有效性.
A full multigrid method is proposed to solve the semilinear elliptic problem by the finite element method based on the combination of multilevel correction method and multigrid method for the linear elliptic problems. In the proposed method, solving the semilinear problem is decomposed into solutions of the linear elliptic problem by the multigrid method,and the semilinear problem which is defined in a very low dimension space. With the help of parallel computing technique, the overfull efficiency can be improved clearly. Furthermore, when the nonlinear term is a polynomial function, an efficient full multigrid method is designed such that the asymptotically computational work is absolutely optimal. One numerical example is provided to validate the efficiency of the proposed method in this paper.
A full multigrid method is proposed to solve the semilinear elliptic problem by the finite element method based on the combination of multilevel correction method and multigrid method for the linear elliptic problems. In the proposed method, solving the semilinear problem is decomposed into solutions of the linear elliptic problem by the multigrid method,and the semilinear problem which is defined in a very low dimension space. With the help of parallel computing technique, the overfull efficiency can be improved clearly. Furthermore, when the nonlinear term is a polynomial function, an efficient full multigrid method is designed such that the asymptotically computational work is absolutely optimal. One numerical example is provided to validate the efficiency of the proposed method in this paper.
引文
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[3] Bramble J H. Multigrid Methods. Pitman Research Notes in Mathematics, Vol. 294, John Wiley and Sons, 1993.
[4] Bramble J H, Pasciak J E. New convergence estimates for multigrid algorithms[J]. Math. Comp.,1987, 49:311-329.
[5] Bramble J H, Zhang X. The Analysis of Multigrid Methods. Handbook of Numerical Analysis,2000, 173-415.
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[7] Brenner S,Scott L. The Mathematical Theory of Finite Element Methods. New York,SpringerVerlag, 1994.
[8] Ciarlet P G.The Finite Element Method for Elliptic Problem. North-holland Amsterdam,1978.
[9]Hackbusch W. Multi-grid Methods and Applications. Springer-Verlag,Berlin, 1985.
[10] Huang Y,Shi Z,Tang T,Xue W. A multilevel successive iteration method for nonlinear elliptic problem[J]. Math. Comp., 2004, 73:525-539.
[11] Jia S, Xie H, Xie M, Xu F. A full multigrid method for monlinear eigenvalue problems[J]. Sci.China Math., 2016, 59:2037-2048.
[12]林群,谢和虎.有限元Aubin-Nitsche技巧新认识及其应用[J].数学的实践与认识,2011, 41(17):247-258.
[13] Lin Q, Xie H. A multi-level correction scheme for eigenvalue problems[J]. Math. Comp., 2015,84(291):71-88.
[14] Lin Q, Xie H, Xu F. Multilevel correction adaptive finite element method for semilinear elliptic equation[J]. Appl. Math., 2015, 60(5):527-550.
[15] Scott L, Zhang S. Higher dimensional non-nested multigrid methods[J]. Math. Comp., 1992, 58:457-466.
[16] Toselli A, Widlund O. Domain Decomposition Methods:Algorithm and Theory. Springer-Verlag,Berlin Heidelberg, 2005.
[17] Shaidurov V V. Multigrid Methods for Finite Elements. Kluwer Academic Publics,Netherlands,1995.
[18] Xie H. A type of multilevel method for the Steklov eigenvalue problem[J]. IMA J. Numer. Anal.,2014, 34:592-608.
[19] Xie H. A multigrid method for eigenvalue problem[J]. J. Comput. Phys., 2014, 274:550-561.
[20] Xie H.非线性特征值问题的多重网格算法.中国科学:数学,2015, 45(8):1193-1204.
[21] Xie H, Xie M. A multigrid method for the ground state solution of Bose-Einstein condensates[J].Commun. Comput. Phys., 2016, 19(3):648-662.
[22] Xu J. Iterative methods by space decomposition and subspace correction[J]. SIAM Review,1992,34(4):581-613.
[23] Xu J. Two-grid discretization techniques for linear and nonlinear PDEs[J]. SIAM J. Numer. Anal.,1996, 33(5):1759-1777.
[24] Xu J. A novel two-grid method for semilinear elliptic equations[J]. SIAM J. Sci. Comput., 1994,15(1):231-237.
[2] Bank R E, Dupont T. An optimal order process for solving finite element equations[J]. Math.Comp., 1981, 36:35-51.
[3] Bramble J H. Multigrid Methods. Pitman Research Notes in Mathematics, Vol. 294, John Wiley and Sons, 1993.
[4] Bramble J H, Pasciak J E. New convergence estimates for multigrid algorithms[J]. Math. Comp.,1987, 49:311-329.
[5] Bramble J H, Zhang X. The Analysis of Multigrid Methods. Handbook of Numerical Analysis,2000, 173-415.
[6] Brandt A, McCormick S,Ruge J. Multigrid methods for differential eigenproblems[J]. SIAM J.Sci. Stat. Comput., 1983, 4(2):244-260.
[7] Brenner S,Scott L. The Mathematical Theory of Finite Element Methods. New York,SpringerVerlag, 1994.
[8] Ciarlet P G.The Finite Element Method for Elliptic Problem. North-holland Amsterdam,1978.
[9]Hackbusch W. Multi-grid Methods and Applications. Springer-Verlag,Berlin, 1985.
[10] Huang Y,Shi Z,Tang T,Xue W. A multilevel successive iteration method for nonlinear elliptic problem[J]. Math. Comp., 2004, 73:525-539.
[11] Jia S, Xie H, Xie M, Xu F. A full multigrid method for monlinear eigenvalue problems[J]. Sci.China Math., 2016, 59:2037-2048.
[12]林群,谢和虎.有限元Aubin-Nitsche技巧新认识及其应用[J].数学的实践与认识,2011, 41(17):247-258.
[13] Lin Q, Xie H. A multi-level correction scheme for eigenvalue problems[J]. Math. Comp., 2015,84(291):71-88.
[14] Lin Q, Xie H, Xu F. Multilevel correction adaptive finite element method for semilinear elliptic equation[J]. Appl. Math., 2015, 60(5):527-550.
[15] Scott L, Zhang S. Higher dimensional non-nested multigrid methods[J]. Math. Comp., 1992, 58:457-466.
[16] Toselli A, Widlund O. Domain Decomposition Methods:Algorithm and Theory. Springer-Verlag,Berlin Heidelberg, 2005.
[17] Shaidurov V V. Multigrid Methods for Finite Elements. Kluwer Academic Publics,Netherlands,1995.
[18] Xie H. A type of multilevel method for the Steklov eigenvalue problem[J]. IMA J. Numer. Anal.,2014, 34:592-608.
[19] Xie H. A multigrid method for eigenvalue problem[J]. J. Comput. Phys., 2014, 274:550-561.
[20] Xie H.非线性特征值问题的多重网格算法.中国科学:数学,2015, 45(8):1193-1204.
[21] Xie H, Xie M. A multigrid method for the ground state solution of Bose-Einstein condensates[J].Commun. Comput. Phys., 2016, 19(3):648-662.
[22] Xu J. Iterative methods by space decomposition and subspace correction[J]. SIAM Review,1992,34(4):581-613.
[23] Xu J. Two-grid discretization techniques for linear and nonlinear PDEs[J]. SIAM J. Numer. Anal.,1996, 33(5):1759-1777.
[24] Xu J. A novel two-grid method for semilinear elliptic equations[J]. SIAM J. Sci. Comput., 1994,15(1):231-237.