非线性Sobolev-Galpern型湿气迁移方程的最低阶非协调混合元收敛性分析
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摘要
对一类非线性Sobolev-Galpern型湿气迁移方程利用非协调线性三角形元和P_0×P_0元,构造一个新的低阶非协调混合元格式,并证明逼近格式解的存在唯一性.同时,在抛弃传统混合元分析的必要工具Ritz投影的前提下,直接利用单元特性,分别得到原始变量u的H1模意义下和中间变量■的L2模意义下的最优误差估计.
Based on the nonconforming linear triangular finite element,the lowest noconforming mixed finite element approximate scheme is established for nonlinear Sobolev-Galpern type equations of moisture migration. The existence and uniqueness of approximation solution are proved. At the same time,without the conventional Ritz projection,the optimal error estimates of exact solution u in H1-norm and intermediate variable ■ in L2-norm are deduced by some special properties of the elements.
Based on the nonconforming linear triangular finite element,the lowest noconforming mixed finite element approximate scheme is established for nonlinear Sobolev-Galpern type equations of moisture migration. The existence and uniqueness of approximation solution are proved. At the same time,without the conventional Ritz projection,the optimal error estimates of exact solution u in H1-norm and intermediate variable ■ in L2-norm are deduced by some special properties of the elements.
引文
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[2]刘亚成,王峰.一类非线性多维Sobolev-Galpern方程[J].应用数学学报,1994,17(4):569-577.
[3]施德明.非线性湿气迁移方程的初边值问题[J].应用数学学报,1990,13(1):31-38.
[4]FORD W H,TING T W. Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations[J]. SIAM J Numer Anal,1974,11(1):155-169.
[5]BONA J L,PRITCHARD W G,SCOOT L R. Numerical schemes for a model for nonlinear dispersive waves[J]. J Comput Phys,1985,60(1):167-186.
[6]QUARTERTERONI A. Fourier spectral methods for pseudo-parabolic equations[J]. Math Comp,1981,36(153):53-63.
[7]周家全,许超,裴丽芳,等.一类非线性Sobolev-Galpern型湿气迁移方程的质量集中非协调有限元方法[J].生物数学学报,2013,28(2):324-330.
[8]裴丽芳,亢金轩,许超.非线性湿气迁移方程Carey非协调元逼近[J].山西大学学报(自然科学版),2011,39(6):81-83.
[9]RAVIART P A,THOMAS J M. A Mixed Finite Element Method for Second Order Elliptic Problems[M]. New York:SpingerVerlag,1977:292-315.
[10]BREZZI F,DOUGLAS J,FORTIN M,et al. Efficient rectangular mixed finite elements in two and three space variables[J].RAIRO,Model Math Anal Numer,1987,21:581-604.
[11]陈绍春,陈红如.二阶椭圆问题新的混合元格式[J].计算数学,2010,32(2):213-218.
[12]史峰,于佳平,李开泰.椭圆方程的一种新型混合有限元格式[J].工程数学学报,2011,28(2):231-237.
[13]石东洋,李明浩.二阶椭圆问题一种新格式的高精度分析[J].应用数学学报,2014,37(1):45-58.
[14]石东洋,张亚东.抛物型方程一个新的非协调混合元超收敛分析与外推[J].计算数学,2013,35(4):337-352.
[15]史艳华,石东洋. Sobolev方程新混合元方法的高精度分析[J].系统科学与数学,2014,34(4):452-463.
[16]罗娟,毛凤梅,王俊俊.非线性S-G型湿气迁移方程的混合元超收敛分析及外推[J].安徽大学学报(自然科学版),2016,40(5):18-23.
[17]罗娟,张厚超,王俊俊.非线性S-G型湿气迁移方程的H1-Galerkin混合元方法[J].山西大学学报(自然科学版),2016,39(2):196-203.
[18]HALE J K. Ordinary Differential Equations[M]. New York:Willey-Inter Science,1969.
[19]Sh I D Y,GUAN H B. A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes[J]. Hokkaido Mathematical Journal 2007,36(4):687-709.