This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error es...This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates.The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element,and the control variable is approximated by piecewise constant functions.The time derivative is discretized by the backward Euler method.Firstly,we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis.Secondly,we derive a priori error estimates for all variables.Thirdly,we present a two-grid scheme and analyze its convergence.In the two-grid scheme,the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.At last,a numerical example is presented to verify the theoretical results.展开更多
In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the ...In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).展开更多
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-...Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.展开更多
基金supported by National Natural Science Foundation of China(No.12571388)the Visiting scholar program of National Natural Science Foundation of China(No.12426616)Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(No.NY223127)。
文摘This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates.The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element,and the control variable is approximated by piecewise constant functions.The time derivative is discretized by the backward Euler method.Firstly,we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis.Secondly,we derive a priori error estimates for all variables.Thirdly,we present a two-grid scheme and analyze its convergence.In the two-grid scheme,the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.At last,a numerical example is presented to verify the theoretical results.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China 10971074+1 种基金the National Basic Research Program under the Grant 2005CB321703Hunan Provincial Innovation Foundation For Postgraduate CX2009B119。
文摘In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).
基金Supported by the National Natural Science Foundation of China Grant (No. 10771124)the Research Fund for Doctoral Program of High Education by State Education Ministry of China (No. 20060422006)+1 种基金the Program for Innovative Research Team in Ludong Universitythe Discipline Construction Fund of Ludong University
文摘Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.
基金Supported by the National Natural Science Fund(10601022)NSF of Inner Mongolia Autonomous Region(200607010106)513 Fund and Youth Science Fund (ND0702) in Inner Mongolia University
