求解带间断系数非线性椭圆问题的两重网格法
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摘要
利用有限差分法离散带间断系数的非线性椭圆问题,针对离散后所得到的非线性方程组,从减少计算量的角度出发,只使用一个辅助的粗层网格空间,并在最细层网格上求解线性校正方程组,构造了两重网格(NETG)法.数值结果表明,新算法在计算量和计算时间方面优于以往的算法.
An efficient two-level method is studied to solve the nonlinear systems which arise from a nonlinear elliptic problem with a jumping coefficient by a finite difference method. Only one auxiliary coarse grid space is needed with a suitable mesh size discretization, and linear correction equations are applied instead of the nonlinear systems on the finest grid space. Numerical results are given to verify the computational cost and efficiency of the proposed approach and the results are compared with the existing numerical methods for solving nonlinear elliptic problems with a jumping coefficient.
An efficient two-level method is studied to solve the nonlinear systems which arise from a nonlinear elliptic problem with a jumping coefficient by a finite difference method. Only one auxiliary coarse grid space is needed with a suitable mesh size discretization, and linear correction equations are applied instead of the nonlinear systems on the finest grid space. Numerical results are given to verify the computational cost and efficiency of the proposed approach and the results are compared with the existing numerical methods for solving nonlinear elliptic problems with a jumping coefficient.
引文
[1] BRANDT A.Multi-Level Adaptive Solutions to Boundary-Value Problems [J].Mathematics of Computation,1977,31(138):333-390.
[2] LI M,LI C L,CUI X Z,et al.Cascadic Multigrid Methods Combined with Sixth Order Compact Scheme for Poisson Equation [J].Numerical Algorithms,2016,71(4):715-727.
[3] 李明,赵金娥.二维椭圆问题的经济外推瀑布多重网格法 [J].西南大学学报(自然科学版),2014,36(7):68-72.
[4] LI M,LI C L.New Cascadic Multigrid Methods for Two-Dimensional Poisson Problem Based on the Fourth-Order Compact Difference Scheme [J].Mathematical Methods in the Applied Sciences,2018,41(3):920-928.
[5] 潘克家,汤井田,胡宏伶,等.直流电阻率法2.5维正演的外推瀑布式多重网格法 [J].地球物理学报,2012,55(8):2769-2778.
[6] TIMMERMANN G.A Cascadic Multigrid Algorithm for Semilinear Elliptic Problems [J].Numerische Mathematik,2000,86(4):717-731.
[7] 邹战勇.半线性椭圆型问题Mortar有限元逼近的瀑布型多重网格法 [J].数学理论与应用,2006,26(1):39-41.
[8] 禹海雄,孙哲.一类半线性椭圆问题的瀑布型多重网格法 [J].湖南大学学报(自然科学版),2011,38(8):79-81.
[9] 禹海雄.几类非线性问题的多重网格解法 [D].长沙:湖南大学,2011.
[10] 陈维翰.用多重网格法改进偏微分方程反问题的广义脉冲谱算法 [J].贵州师范大学学报(自然科学版),1995,13(4):1-6.
[2] LI M,LI C L,CUI X Z,et al.Cascadic Multigrid Methods Combined with Sixth Order Compact Scheme for Poisson Equation [J].Numerical Algorithms,2016,71(4):715-727.
[3] 李明,赵金娥.二维椭圆问题的经济外推瀑布多重网格法 [J].西南大学学报(自然科学版),2014,36(7):68-72.
[4] LI M,LI C L.New Cascadic Multigrid Methods for Two-Dimensional Poisson Problem Based on the Fourth-Order Compact Difference Scheme [J].Mathematical Methods in the Applied Sciences,2018,41(3):920-928.
[5] 潘克家,汤井田,胡宏伶,等.直流电阻率法2.5维正演的外推瀑布式多重网格法 [J].地球物理学报,2012,55(8):2769-2778.
[6] TIMMERMANN G.A Cascadic Multigrid Algorithm for Semilinear Elliptic Problems [J].Numerische Mathematik,2000,86(4):717-731.
[7] 邹战勇.半线性椭圆型问题Mortar有限元逼近的瀑布型多重网格法 [J].数学理论与应用,2006,26(1):39-41.
[8] 禹海雄,孙哲.一类半线性椭圆问题的瀑布型多重网格法 [J].湖南大学学报(自然科学版),2011,38(8):79-81.
[9] 禹海雄.几类非线性问题的多重网格解法 [D].长沙:湖南大学,2011.
[10] 陈维翰.用多重网格法改进偏微分方程反问题的广义脉冲谱算法 [J].贵州师范大学学报(自然科学版),1995,13(4):1-6.