Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonl...Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since （Vh, Mh） does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of （Vh, Mh） is established. Under these conditions, we obtain the H1 and L2 error estimates for the numerical solutions.展开更多
In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th...In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.展开更多
Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these m...Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.展开更多
In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic id...In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.展开更多
In this paper,we propose a local discontinuous Galerkin(LDG)method for themulti-dimensional stochastic Cahn-Hilliard type equation in a general form,which involves second-order derivative Du in the multiplicative nois...In this paper,we propose a local discontinuous Galerkin(LDG)method for themulti-dimensional stochastic Cahn-Hilliard type equation in a general form,which involves second-order derivative Du in the multiplicative noise.The stability of our scheme is proved for arbitrary polygonal domain with triangular meshes.We get the sub-optimal error estimate O(h^(k))if the Cartesian meshes with Q^(k) elements are used.Numerical examples are given to display the performance of the LDG method.展开更多
We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties.An important feature of the methods is that the press...We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties.An important feature of the methods is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations.Although the stability of the method has been established,for the homogeneous Stokes equations,the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space.This makes it much simpler and more attractive.The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.展开更多
基金supported by the National Natural Science Foundation of China(10901122)Zhejiang Provincial Natural Science Foundation (Y6090108)supported by the National Natural Science Foundation of China(10971165)
文摘Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since （Vh, Mh） does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of （Vh, Mh） is established. Under these conditions, we obtain the H1 and L2 error estimates for the numerical solutions.
文摘In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.
基金The research of the first author is support by NSFC grant 10601055,FANEDD of CAS and SRF for ROCS SEMThe research of the second author is supported by NSF grant DMS-0809086 and DOE grant DE-FG02-08ER25863.
文摘Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11601241,11671199,11571290 and 11672082)Natural Science Foundation of Jiangsu Province(Grant No.BK20160877)+1 种基金ARO(Grant No.W911NF-15-1-0226)National Science Foundation of USA(Grant No.DMS-1719410)
文摘In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.
基金funded by National Key R&D Program of China(Grant Nos.2020YFA0711900,2020YFA0711902)funded by China National Postdoctoral Program for Innovative Talents(Grant No.BX20200096)China Postdoctoral Science Foundation(Grant No.2021M690703).
文摘In this paper,we propose a local discontinuous Galerkin(LDG)method for themulti-dimensional stochastic Cahn-Hilliard type equation in a general form,which involves second-order derivative Du in the multiplicative noise.The stability of our scheme is proved for arbitrary polygonal domain with triangular meshes.We get the sub-optimal error estimate O(h^(k))if the Cartesian meshes with Q^(k) elements are used.Numerical examples are given to display the performance of the LDG method.
文摘We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties.An important feature of the methods is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations.Although the stability of the method has been established,for the homogeneous Stokes equations,the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space.This makes it much simpler and more attractive.The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations.
