The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete mode...The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.展开更多
In this paper,we present a Hessian recovery based linear finite element method to simulate the molecular beam epitaxy growth model with slope selection.For the time discretization,we apply a first-order convex splitti...In this paper,we present a Hessian recovery based linear finite element method to simulate the molecular beam epitaxy growth model with slope selection.For the time discretization,we apply a first-order convex splitting method and secondorder Crank-Nicolson scheme.For the space discretization,we utilize the Hessian recovery operator to approximate second-order derivatives of a C^(0)linear finite element function and hence the weak formulation of the fourth-order differential operator can be discretized in the linear finite element space.The energy-decay property of our proposed fully discrete schemes is rigorously proved.The robustness and the optimal-order convergence of the proposed algorithm are numerically verified.In a large spatial domain for a long period,we simulate coarsening dynamics,where 1/3-power-law is observed.展开更多
In this paper,we introduce a new stabilized continuous linear element method for solving biharmonic problems.Leveraging the gradient recovery operator,we reconstruct the discrete Hessian for piecewise continuous linea...In this paper,we introduce a new stabilized continuous linear element method for solving biharmonic problems.Leveraging the gradient recovery operator,we reconstruct the discrete Hessian for piecewise continuous linear functions.By adding a stability term to the discrete bilinear form,we bypass the need for the discrete Poincaréinequality.We employ Nitsche's method for weakly enforcing boundary conditions.We establish well-posedness of the solution and derive optimal error estimates in energy and L^(2) norms.Numerical results are provided to validate our theoretical findings.展开更多
This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average inter...This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.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
Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of or...Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h^(1+min){α,1}) is established for both the displacement approximation in H^1-norm and the stress approximation in L^2-norm under a mesh assumption, where α > 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.展开更多
The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete Hi-norm and global extrapolation in discrete L2-nor...The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete Hi-norm and global extrapolation in discrete L2-norm are proved. Based on these global estimates the conjugate gradient method （CG） is effective, which is applied to extrapolation cascadic multigrid method （EXCMG）. The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.展开更多
基金the National Nuclear Security Administration of the U.S.Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396the DOE Office of Science Advanced Scientific Computing Research(ASCR)Program in Applied Mathematics Research.The first author has been supported in part by the Czech Ministry of Education projects MSM 6840770022 and LC06052(Necas Center for Mathematical Modeling).
文摘The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.
基金supported by General Scientific Research Projects of Zhejiang Education Department(No.Y202147013)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-Sen University(No.2021008)+1 种基金supported in part by NSFC Grant(No.12071496)Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(No.2020B1212060032)。
文摘In this paper,we present a Hessian recovery based linear finite element method to simulate the molecular beam epitaxy growth model with slope selection.For the time discretization,we apply a first-order convex splitting method and secondorder Crank-Nicolson scheme.For the space discretization,we utilize the Hessian recovery operator to approximate second-order derivatives of a C^(0)linear finite element function and hence the weak formulation of the fourth-order differential operator can be discretized in the linear finite element space.The energy-decay property of our proposed fully discrete schemes is rigorously proved.The robustness and the optimal-order convergence of the proposed algorithm are numerically verified.In a large spatial domain for a long period,we simulate coarsening dynamics,where 1/3-power-law is observed.
文摘In this paper,we introduce a new stabilized continuous linear element method for solving biharmonic problems.Leveraging the gradient recovery operator,we reconstruct the discrete Hessian for piecewise continuous linear functions.By adding a stability term to the discrete bilinear form,we bypass the need for the discrete Poincaréinequality.We employ Nitsche's method for weakly enforcing boundary conditions.We establish well-posedness of the solution and derive optimal error estimates in energy and L^(2) norms.Numerical results are provided to validate our theoretical findings.
文摘This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.
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supported by National Natural Science Foundation of China (Grant No. 11171239)Major Research Plan of National Natural Science Foundation of China (Grant No. 91430105)
文摘Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h^(1+min){α,1}) is established for both the displacement approximation in H^1-norm and the stress approximation in L^2-norm under a mesh assumption, where α > 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.
基金supported by National Natural Science Foundation of China(Grant Nos.1130117611071067 and 11226332)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036)the Construct Program of the Key Discipline in Hunan Province
文摘The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete Hi-norm and global extrapolation in discrete L2-norm are proved. Based on these global estimates the conjugate gradient method （CG） is effective, which is applied to extrapolation cascadic multigrid method （EXCMG）. The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.
