Abstract.In this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear...Abstract.In this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic form.We use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time discretization.Both mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method.Numerical examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.展开更多
This paper considers the Legendre Galerkin spectral approximation for the unconstralnea optimal control problems. The authors derive a posteriori error estimate for the spectral approximation scheme of optimal control...This paper considers the Legendre Galerkin spectral approximation for the unconstralnea optimal control problems. The authors derive a posteriori error estimate for the spectral approximation scheme of optimal control problem. By choosing the appropriate basis functions, the stiff matrix of the discretization equations is sparse. And the authors use the Fast Legendre Transform to improve the efficiency of this method. Two numerical experiments demonstrating our theoretical results are presented.展开更多
We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation.The main idea is to replace the gradient operator▽on linear finite element space by G(▽)in the wea...We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation.The main idea is to replace the gradient operator▽on linear finite element space by G(▽)in the weak formulation of the biharmonic equation,where G is the recovery operator which recovers the piecewise constant function into the linear finite element space.By operator G,Laplace operator△is replaced by▽·G(▽).Furthermore,the boundary condition on normal derivative▽u-n is treated by the boundary penalty method.The explicit matrix expression of the proposed method is also introduced.Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.展开更多
We propose some new weighted averaging methods for gradient recovery,and present analytical and numerical investigation on the performance of these weighted averaging methods.It is shown analytically that the harmonic...We propose some new weighted averaging methods for gradient recovery,and present analytical and numerical investigation on the performance of these weighted averaging methods.It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension.Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.展开更多
基金Yi’s research was partially supported by NSFC Project(No.12071400)China’s National Key R&D Programs(No.2020YFA0713500)+2 种基金Hunan Provincial NSF Project Yi’s research was partially supported by NSFC Project(No.12071400)China’s National Key R&D Programs(No.2020YFA0713500)Hunan Provincial NSF Project。
文摘Abstract.In this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic form.We use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time discretization.Both mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method.Numerical examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.
基金supported by the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)the National Natural Science Foundation of China under Grant No.10971074
文摘This paper considers the Legendre Galerkin spectral approximation for the unconstralnea optimal control problems. The authors derive a posteriori error estimate for the spectral approximation scheme of optimal control problem. By choosing the appropriate basis functions, the stiff matrix of the discretization equations is sparse. And the authors use the Fast Legendre Transform to improve the efficiency of this method. Two numerical experiments demonstrating our theoretical results are presented.
文摘We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation.The main idea is to replace the gradient operator▽on linear finite element space by G(▽)in the weak formulation of the biharmonic equation,where G is the recovery operator which recovers the piecewise constant function into the linear finite element space.By operator G,Laplace operator△is replaced by▽·G(▽).Furthermore,the boundary condition on normal derivative▽u-n is treated by the boundary penalty method.The explicit matrix expression of the proposed method is also introduced.Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.
基金The first author is supported by supported in part by the NSFC Key Project 11031006Hunan Provincial NSF project 10JJ7001+1 种基金The third author is supported by Hunan Education Department Key Project 10A117Hunan Provincial Innovation Foundation For Postgraduate(Grant No.S2008yjscx05)。
文摘We propose some new weighted averaging methods for gradient recovery,and present analytical and numerical investigation on the performance of these weighted averaging methods.It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension.Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.
