非线性抛物方程混合有限元方法的高精度分析
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摘要
采用双线性元及零阶Raviart-Thomas元(Q_(11)+Q_(10)×Q_(01))对非线性抛物方程讨论了一种H~1-Galerkin混合有限元方法.提出一个线性化的二阶格式,利用数学归纳法有技巧的导出了原始变量u在H~1(Ω)模意义下及流量■=▽u在L~2(Ω)模意义下的O(h~2+τ~2)阶超逼近性质.引入一个有关初始点的时间离散方程,并利用其得到了▽·■在L~2(Ω)模意义下的O(h~2+τ~2)阶的超逼近结果.同时利用插值后处理技巧得到整体超收敛.最后,数值算例结果验证了理论分析(其中,h是剖分参数,τ是时间步长).
An H~1-Galerkin mixed finite element method is discussed for nonlinear parabolic equations with the bilinear element and the zero-order Raviart-Thomas element(Q_(11)+Q_(10)×Q_(01)).A linearized second order fully-discrete scheme is proposed. The superclose results with O(h~2 +τ~2) of original variant u in H~1-norm and flux variant ■ in L2-norm are derived technically. A time semi-discrete equation at the starting point is introduced and the superclose property of ▽·■in L2-norm is reduced. Furthermore, the corresponding global superconvergence results are obtained by the interpolated postprocessing technique. At last,numerical results are presented to illustrate the feasibility of the proposed method(Here, h is the subdivision parameter, and τ, the time step).
An H~1-Galerkin mixed finite element method is discussed for nonlinear parabolic equations with the bilinear element and the zero-order Raviart-Thomas element(Q_(11)+Q_(10)×Q_(01)).A linearized second order fully-discrete scheme is proposed. The superclose results with O(h~2 +τ~2) of original variant u in H~1-norm and flux variant ■ in L2-norm are derived technically. A time semi-discrete equation at the starting point is introduced and the superclose property of ▽·■in L2-norm is reduced. Furthermore, the corresponding global superconvergence results are obtained by the interpolated postprocessing technique. At last,numerical results are presented to illustrate the feasibility of the proposed method(Here, h is the subdivision parameter, and τ, the time step).
引文
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[11] Pani A K. An H~1-Galerkin mixed finite element methods for parabolic partial differential equatios[J]. SIAM J. Numer. Anal., 1998, 35(2):712-727.
[12] Guo L, Chen H. H~1-Galerkin mixed finite element method for the regularized long wave equation[J]. Computing, 2006, 77:205-221.
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[14] Liu Y, Li H. H~1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl.Math. Comput., 2009, 212(2):446-457.
[15]石东洋,史艳华.半线性伪双曲方程最低阶的H~1-Galerkin混合元方法[J].系统科学与数学,2015,35(5):514-526.
[16] Pani A K, Fairweather G. An H~1-galerkin mixed finite element method for an evolution equation with a positive-type memory term[J]. SIAM J. Numer. Anal., 2002, 40(4):1475-1490.
[17] Pani A K. H~1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal., 2002, 22:231-252.
[18] Chen F X. Crank-Nicolson fully discrete H~1-Galerkin mixed finite element approximation of one nonlinear integro-differential model[J]. Abstr. Appl. Anal., 2014, Article ID 534902, 8 pages.
[19] Shi D Y, Liao X, Tang Q L. Highly efficient H~1-Galerkin mixed finite element method(MFEM)for parabolic integro-differential equatios[J]. Appl. Math. Mech.-Engl. Ed., 2014, 35(7):897-912.
[20] Zhang Y D, Shi D Y. Superconvergence of an H~1-Galerkin nonconforming mixed finite element method for a parabolic equation[J]. Comput. Math. Appl., 2013, 66(11):2362-2375.
[21] Sun T J, Ma K Y. Domain decomposition procedures combined with H~1-Galerkin mixed finite element method for parabolic equation[J]. J. Comput. Appl. Math., 2014, 267:33-48.
[22] Shi D Y, Wang J J. Superconvergence analysis of an H~1-Galerkin mixed finite element method for Sobolev equations[J]. Comput. Math. Appl., 2016, 72(6):1590-1602.
[23] Lin Q, Lin J F. Finite element methods:accuracy and improvement. Science press, Beijing(2006).
[2] Chatzipantelidis P, Lazarov R D, Thomee V. Error estimates for the finite volume element methodfor parobolic equations in convex polygonal domains[J]. Numer. Methods Partial Differential Equations, 2003, 20:650-674.
[3] Wang T K. Alternating direction finite volume element methods for 2D parabolic partial differential equations[J]. Numer. Methods Partial Differential Equations., 2008, 24(1):24-40.
[4] Sinha R K, Ewing R E, Lazarov R D. Some new error estimates of a semidicrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data[J]. SIAM J. Numer. Anal., 2008, 43(6):2320-2344.
[5] Li Q H, Wang J P. Weak Galerkin finite element methods for parabolic equations. Numer[J].Methods Partial Differential Equations, 2013, 29(6):2004-2024.
[6] Thomée V. Galerkin finite element methods for parabolic problems. Springer series in computational mathematics, Sweden(2000).
[7] Luskin M. A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions[J]. SIAM J. Numer. Anal., 1979, 16(2):284-299.
[8] Shi D Y, Wang J J. Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation[J]. Comput. Math. Appl., 2017, 294:216-226.
[9] Shi D Y, Wang J J, Yan F N. Unconditional superconvergence analysis for nonlinear parabolic equation with EQ_1~(rot)nonconforming finite element[J]. J. Sci. Comput., 2017, 70(1):85-111.
[10] Li D F, Wang J L. Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system[J]. J. Sci. Comput., 2017:1-24.
[11] Pani A K. An H~1-Galerkin mixed finite element methods for parabolic partial differential equatios[J]. SIAM J. Numer. Anal., 1998, 35(2):712-727.
[12] Guo L, Chen H. H~1-Galerkin mixed finite element method for the regularized long wave equation[J]. Computing, 2006, 77:205-221.
[13] Pani A K, Sinha R K, OTTA A K. An H~1-Galerkin mixed method for second order hyperbolic equations[J]. Int. J. Numer. Anal. Model., 2004, 1(2):111-130.
[14] Liu Y, Li H. H~1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl.Math. Comput., 2009, 212(2):446-457.
[15]石东洋,史艳华.半线性伪双曲方程最低阶的H~1-Galerkin混合元方法[J].系统科学与数学,2015,35(5):514-526.
[16] Pani A K, Fairweather G. An H~1-galerkin mixed finite element method for an evolution equation with a positive-type memory term[J]. SIAM J. Numer. Anal., 2002, 40(4):1475-1490.
[17] Pani A K. H~1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal., 2002, 22:231-252.
[18] Chen F X. Crank-Nicolson fully discrete H~1-Galerkin mixed finite element approximation of one nonlinear integro-differential model[J]. Abstr. Appl. Anal., 2014, Article ID 534902, 8 pages.
[19] Shi D Y, Liao X, Tang Q L. Highly efficient H~1-Galerkin mixed finite element method(MFEM)for parabolic integro-differential equatios[J]. Appl. Math. Mech.-Engl. Ed., 2014, 35(7):897-912.
[20] Zhang Y D, Shi D Y. Superconvergence of an H~1-Galerkin nonconforming mixed finite element method for a parabolic equation[J]. Comput. Math. Appl., 2013, 66(11):2362-2375.
[21] Sun T J, Ma K Y. Domain decomposition procedures combined with H~1-Galerkin mixed finite element method for parabolic equation[J]. J. Comput. Appl. Math., 2014, 267:33-48.
[22] Shi D Y, Wang J J. Superconvergence analysis of an H~1-Galerkin mixed finite element method for Sobolev equations[J]. Comput. Math. Appl., 2016, 72(6):1590-1602.
[23] Lin Q, Lin J F. Finite element methods:accuracy and improvement. Science press, Beijing(2006).