非线性方程的混合有限元研究
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摘要
本文针对不同类型的非线性方程(包括磁流体动力学方程、带有阻尼项的Stokes方程、对流扩散方程、反应-扩散方程及非线性抛物方程等),分别从非协调有限元方法、最小二乘有限元方法、误差常数的精细估计、新的混合变分形式下的二重网格法、变网格法等不同角度出发,对混合有限元方法的构造,收敛性分析、超逼近和超收敛等方而进行深入系统的探讨.
首先考虑了一类低阶非协调单元(包括四而体元和六而体元),用于逼近一个三维的定常不可压、完全耦合的非线性磁流体动力学方程,在磁场分别属于H1(Ω)3和H(curl;Ω)时,采用非协调混合有限元法来分析文献[14]和[27]中的方程,证明了离散问题解的存在惟一性并给出了相应未知量的最优误差估计.而且采用一种新的方法证明了离散的Poincare-Friedrichs不等式.
其次用带约束的非协调旋转Q1元和分片常数元来逼近定常的、不可压带有阻尼项的Stokes方程的速度和压力.证明了逼近解的存在惟一性.再利用精确解和逼近解的先验估计,并恰当选择方程中出现的参数α,ν和r,得到了最优误差估计及超逼近结果.最后,通过插值后处理技术,导出了速度的H1-模和压力L2-模的O(h2)阶的整体超收敛.
再次研究了对流扩散方程的最小二乘非协调有限元格式及其两种修正格式,用矩形EQ1rot元和零阶R-T元分别来逼近位移和应力,借助于单元本身的特殊性质,给出了逼近问题解的存在惟一性,得到了位移H1-模和应力H(div)-模的O(h)阶的误差估计.同时,在直角三角形网格下,采用最小二乘有限元法,用零阶的R-T元和P1元去离散该方程,给出了应力的H(div)-模及位移的H1-模的误差常数的精细估计,并给出了数值算例验证了理论分析的正确性.
而后构造了一种新的混合变分形式,用最低阶矩形协调混合元去逼近半线性反应扩散方程,利用椭圆投影的特殊性质,得到了新的混合变分全离散形式下的未知量L2-模的误差收敛阶为O(△t+h2)比[136]传统的混合变分形式下的误差提高了一阶.随后,采用二重网格算法,迭代两步得到了网格比H=O(h1/3)及收敛阶O(Δt+h+H3)的结果,这一结果文献[136]需迭代三步才能达到.而本章迭代三步后,得到了网格比是H=O(h2/9),收敛阶为O(Δt+h+H9/2)和[136]相比,达到了迭代同样步数可以增加网格比而减少计算量的目的.
最后利用非协调EQ1rot元和零阶R-T元,对一类非线性抛物方程构造了一种新的混合变网格格式.根据该单元相容误差在能量模意义下比插值误差高一阶的特殊性质,给出了收敛性分析并得到了最优阶误差估计.本章的结果可推广到协调混合有限元逼近的任意收敛阶情形.
In this thesis, we consider some kinds of nonlinear equations (including magne-tohydrodynamics (MHD) equations、Stokes equations with damped term、convection-diffusion equations、reaction-diffusion equations and nonlinear parabolic equations, etc), study the nonconforming finite element methods, the least-square nonconform-ing finite element methods, sharp estimates of error constants and two-grids and moving grids for new mixed variational forms, etc, from different points of view and give deep and comprehensive investigations on the construction of the mixed finite element methods、convergence analysis、superclose and supercorivergence, etc.
Firstly, a family of low order nonconforming elements (including tetrahedra el-ements and hexahedra elements) are used to approximate a nonlinear, fully coupled stationary incompressible MHD equation in 3D. When the magnetic field belongs to H1(Ω)3 and H(curl;Ω), the nonconforming mixed finite element methods are ana-lyzed for the MHD equations discussed in [14] and [27], respectively. The existence and uniqueness of the approximate solutions are proved and the optimal error es-timates about the corresponding unknown variables are given. Furthermore, a new approach is adopted to prove the discrete Poincare-Friedrichs inequality.
Secondly, we apply the constrained nonconforming rotated Q1 element and the piecewise constant element to approximate the velocity and the pressure for the stationary.. impressible Stokes equations with damped term, respectively. The existence and uniqueness of the approximated solutions are proved. Employing the prior estimates of the exact and approximate solutions and choosing the appropriate parameters a, v and r, the optimal error estimates and the superclose results are derived. Finally, the O(h2) global superconvergence in H1-norm for the velocity and L2-norm for the pressure is obtained by use of a postprocessing technique.
Thirdly, we analyze the least-square nonconforming finite element scheme and its two modified forms for the convection-diffusion equations. The rectangular EQ1rot element and zero order R-T element are used to approximate to the displacement and the stress. The existence and uniqueness of the approximate solutions are proved by means of some special properties of the elements. The O(h) order error estimates for the stress in H(div)-norm and the displacement in broken H1-norm are derived. At the same times, we use zero order R-T element and P1 triangular element to discrete the equations by the least-square finite element methods, we get sharp estimations of error constants in H(div)-norm for the stress and L2-norm for the displacement under right angled triangular meshes. Furthermore, the corresponding numerical experiments are carried out to verify the theoretical analysis.
Then, we construct a new mixed variational form for the semilinear reaction-diffusion equations by lowest rectangular conforming mixed finite elements. The error convergence order of O(Δt+h2) in L2-norm about the corresponding unknown functions are derived in the fully-discrete schemes by use of the special properties of elliptic projections and one order is improved than that of [136] using the traditional mixed variational form. Based on two grids algorithms, we derive the meshes ratio H= O(h1/3) and the convergence order O(Δt+h+H3) under the condition of two steps iteration, in order to obtain the results, the literature [136] need iterative three steps. The meshes ratio H= O(h2/9) and the convergence order O(Δt+h+H9/2) are obtained in this chapter under the condition of three steps iteration. Thus, we achieve the purpose of iterating the same step number, increasing the meshes ratio and reducing computation costs.
Finally, we use the nonconforming EQ1rot clement and zero order R-T element and construct a new mixed scheme for nonlinear parabolic equations with moving grids. By use of the special property of the element, i.e., its consistency error is one order higher than the interpolation error in the broken energy norm, the convergence analysis is presented and the optimal order error estimates are derived. The results of this chapter can be extended to any convergence order by conforming mixed finite element approximations.
首先考虑了一类低阶非协调单元(包括四而体元和六而体元),用于逼近一个三维的定常不可压、完全耦合的非线性磁流体动力学方程,在磁场分别属于H1(Ω)3和H(curl;Ω)时,采用非协调混合有限元法来分析文献[14]和[27]中的方程,证明了离散问题解的存在惟一性并给出了相应未知量的最优误差估计.而且采用一种新的方法证明了离散的Poincare-Friedrichs不等式.
其次用带约束的非协调旋转Q1元和分片常数元来逼近定常的、不可压带有阻尼项的Stokes方程的速度和压力.证明了逼近解的存在惟一性.再利用精确解和逼近解的先验估计,并恰当选择方程中出现的参数α,ν和r,得到了最优误差估计及超逼近结果.最后,通过插值后处理技术,导出了速度的H1-模和压力L2-模的O(h2)阶的整体超收敛.
再次研究了对流扩散方程的最小二乘非协调有限元格式及其两种修正格式,用矩形EQ1rot元和零阶R-T元分别来逼近位移和应力,借助于单元本身的特殊性质,给出了逼近问题解的存在惟一性,得到了位移H1-模和应力H(div)-模的O(h)阶的误差估计.同时,在直角三角形网格下,采用最小二乘有限元法,用零阶的R-T元和P1元去离散该方程,给出了应力的H(div)-模及位移的H1-模的误差常数的精细估计,并给出了数值算例验证了理论分析的正确性.
而后构造了一种新的混合变分形式,用最低阶矩形协调混合元去逼近半线性反应扩散方程,利用椭圆投影的特殊性质,得到了新的混合变分全离散形式下的未知量L2-模的误差收敛阶为O(△t+h2)比[136]传统的混合变分形式下的误差提高了一阶.随后,采用二重网格算法,迭代两步得到了网格比H=O(h1/3)及收敛阶O(Δt+h+H3)的结果,这一结果文献[136]需迭代三步才能达到.而本章迭代三步后,得到了网格比是H=O(h2/9),收敛阶为O(Δt+h+H9/2)和[136]相比,达到了迭代同样步数可以增加网格比而减少计算量的目的.
最后利用非协调EQ1rot元和零阶R-T元,对一类非线性抛物方程构造了一种新的混合变网格格式.根据该单元相容误差在能量模意义下比插值误差高一阶的特殊性质,给出了收敛性分析并得到了最优阶误差估计.本章的结果可推广到协调混合有限元逼近的任意收敛阶情形.
In this thesis, we consider some kinds of nonlinear equations (including magne-tohydrodynamics (MHD) equations、Stokes equations with damped term、convection-diffusion equations、reaction-diffusion equations and nonlinear parabolic equations, etc), study the nonconforming finite element methods, the least-square nonconform-ing finite element methods, sharp estimates of error constants and two-grids and moving grids for new mixed variational forms, etc, from different points of view and give deep and comprehensive investigations on the construction of the mixed finite element methods、convergence analysis、superclose and supercorivergence, etc.
Firstly, a family of low order nonconforming elements (including tetrahedra el-ements and hexahedra elements) are used to approximate a nonlinear, fully coupled stationary incompressible MHD equation in 3D. When the magnetic field belongs to H1(Ω)3 and H(curl;Ω), the nonconforming mixed finite element methods are ana-lyzed for the MHD equations discussed in [14] and [27], respectively. The existence and uniqueness of the approximate solutions are proved and the optimal error es-timates about the corresponding unknown variables are given. Furthermore, a new approach is adopted to prove the discrete Poincare-Friedrichs inequality.
Secondly, we apply the constrained nonconforming rotated Q1 element and the piecewise constant element to approximate the velocity and the pressure for the stationary.. impressible Stokes equations with damped term, respectively. The existence and uniqueness of the approximated solutions are proved. Employing the prior estimates of the exact and approximate solutions and choosing the appropriate parameters a, v and r, the optimal error estimates and the superclose results are derived. Finally, the O(h2) global superconvergence in H1-norm for the velocity and L2-norm for the pressure is obtained by use of a postprocessing technique.
Thirdly, we analyze the least-square nonconforming finite element scheme and its two modified forms for the convection-diffusion equations. The rectangular EQ1rot element and zero order R-T element are used to approximate to the displacement and the stress. The existence and uniqueness of the approximate solutions are proved by means of some special properties of the elements. The O(h) order error estimates for the stress in H(div)-norm and the displacement in broken H1-norm are derived. At the same times, we use zero order R-T element and P1 triangular element to discrete the equations by the least-square finite element methods, we get sharp estimations of error constants in H(div)-norm for the stress and L2-norm for the displacement under right angled triangular meshes. Furthermore, the corresponding numerical experiments are carried out to verify the theoretical analysis.
Then, we construct a new mixed variational form for the semilinear reaction-diffusion equations by lowest rectangular conforming mixed finite elements. The error convergence order of O(Δt+h2) in L2-norm about the corresponding unknown functions are derived in the fully-discrete schemes by use of the special properties of elliptic projections and one order is improved than that of [136] using the traditional mixed variational form. Based on two grids algorithms, we derive the meshes ratio H= O(h1/3) and the convergence order O(Δt+h+H3) under the condition of two steps iteration, in order to obtain the results, the literature [136] need iterative three steps. The meshes ratio H= O(h2/9) and the convergence order O(Δt+h+H9/2) are obtained in this chapter under the condition of three steps iteration. Thus, we achieve the purpose of iterating the same step number, increasing the meshes ratio and reducing computation costs.
Finally, we use the nonconforming EQ1rot clement and zero order R-T element and construct a new mixed scheme for nonlinear parabolic equations with moving grids. By use of the special property of the element, i.e., its consistency error is one order higher than the interpolation error in the broken energy norm, the convergence analysis is presented and the optimal order error estimates are derived. The results of this chapter can be extended to any convergence order by conforming mixed finite element approximations.
引文
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