Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal ...Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.展开更多
In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure ...In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^(∞)-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.展开更多
In this paper,the transient Navier-Stokes equations with damping are considered.Firstly,the semi-discrete scheme is discussed and optimal error estimates are derived.Secondly,a linearized backward Euler scheme is prop...In this paper,the transient Navier-Stokes equations with damping are considered.Firstly,the semi-discrete scheme is discussed and optimal error estimates are derived.Secondly,a linearized backward Euler scheme is proposed.By the error split technique,the Stokes operator and the H^(-1)-norm estimate,unconditional optimal error estimates for the velocity in the norms L^(∞)(L^(2)) and L^(∞)(H^(1)),and the pressure in the norm L^(∞)(L^(2))are deduced.Finally,two numerical examples are provided to confirm the theoretical analysis.展开更多
An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same t...An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.展开更多
In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By...In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.展开更多
We examine a simple averaging formula for the gradieni of linear finite elemelitsin Rd whose interpolation order in the Lq-norm is O(h2) for d < 2q and nonuniformtriangulations. For elliptic problems in R2 we deriv...We examine a simple averaging formula for the gradieni of linear finite elemelitsin Rd whose interpolation order in the Lq-norm is O(h2) for d < 2q and nonuniformtriangulations. For elliptic problems in R2 we derive an interior superconvergencefor the averaged gradient over quasiuniform triangulations. Local error estimatesup to a regular part of the boundary and the effect of numerical integration arealso investigated.展开更多
The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error est...The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.展开更多
The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the ...The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.展开更多
This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this elem...This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.展开更多
In this paper, unconventional quasi-conforming finite element approximation for a fourth order variational inequality with displacement obstacle is considered, the optimal order of error estimate O(h) is obtained whic...In this paper, unconventional quasi-conforming finite element approximation for a fourth order variational inequality with displacement obstacle is considered, the optimal order of error estimate O(h) is obtained which is as same as that of the conventional finite elements.展开更多
A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are ...A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.展开更多
In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the...In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.展开更多
In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not...In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving waves.By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are shown.To derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are used.This,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+1.Numerical experiments are given to demonstrate the theoretical results.展开更多
In this paper, a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as ...In this paper, a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as approximation space for the velocity and the linear element for the pressure. The convergence analysis is presented and optimal error estimates of both broken H^1-norm and L^2-norm for velocity as well as the L^2-norm for the pressure are derived.展开更多
In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt ...In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.展开更多
In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is take...In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.展开更多
In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the ...In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition,which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure.For the diffusive term,a pair of generalized alternating fluxes is considered.By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms,we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems.Numerical experiments including long time simulations,different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.展开更多
Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both spac...Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.展开更多
The focus of this paper is on a linearized backward differential formula(BDF)scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations(KGSEs)with damping mechanism.Optimal error estimates and ...The focus of this paper is on a linearized backward differential formula(BDF)scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations(KGSEs)with damping mechanism.Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme.The proof consists of three ingredients.First,a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms.Second,optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms.Third,by virtue of the relationship between the Ritz projection and the interpolation,as well as a so-called"lifting"technique,the superconvergence behavior of order O(h^(2)+τ^(2))in H^(1)-norm for the original variables are deduced.Finally,a numerical experiment is conducted to confirm our theoretical analysis.Here,h is the spatial subdivision parameter,andτis the time step.展开更多
A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the full...A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.展开更多
基金Subsidized by NSFC(11571274 and 11171269)the Ph.D.Programs Foundation of Ministry of Education of China(20110201110027)
文摘Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
基金supported by National Natural Science Foundation of China(No.11671369)the Doctoral Starting Foundation of Zhengzhou University of Aeronautics(No.63020390).
文摘In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^(∞)-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.
基金supported by Fundamental Research Funds for the Henan Provincial Colleges and Universities(No.20A110002).
文摘In this paper,the transient Navier-Stokes equations with damping are considered.Firstly,the semi-discrete scheme is discussed and optimal error estimates are derived.Secondly,a linearized backward Euler scheme is proposed.By the error split technique,the Stokes operator and the H^(-1)-norm estimate,unconditional optimal error estimates for the velocity in the norms L^(∞)(L^(2)) and L^(∞)(H^(1)),and the pressure in the norm L^(∞)(L^(2))are deduced.Finally,two numerical examples are provided to confirm the theoretical analysis.
基金Supported by the National Natural Science Foundation of China (10601022)Natural Science Foundation of Inner Mongolia Autonomous Region (200607010106)Youth Science Foundation of Inner Mongolia University(ND0702)
文摘An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.
基金supported by the NSF of China(Grant Nos.11801527,11701522,11771163,12011530058,11671160,1191101330)by the China Postdoctoral Science Foundation(Grant Nos.2018M632791,2019M662506).
文摘In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.
文摘We examine a simple averaging formula for the gradieni of linear finite elemelitsin Rd whose interpolation order in the Lq-norm is O(h2) for d < 2q and nonuniformtriangulations. For elliptic problems in R2 we derive an interior superconvergencefor the averaged gradient over quasiuniform triangulations. Local error estimatesup to a regular part of the boundary and the effect of numerical integration arealso investigated.
基金The research is supported by NSF of China (10371113 10471133)
文摘The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.
基金supported by the National Natural Science Fundation of China (No. 11061021)the Science Research of Inner Mongolia Advanced Education (Nos. NJ10006, NJ10016, and NJZZ12011)the National Science Foundation of Inner Mongolia (Nos. 2011BS0102 and 2012MS0106)
文摘The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.
基金Supported by the National Natural Science Foundation of China(10371113,10671184)
文摘This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.
基金This research is supported by National Natural Science Foundation of China (10171092), Foundation of Oversea Scholar of China, Project of Creative Engineering of Henan Province and Natural Science Foundation of Henan Province of China.
文摘In this paper, unconventional quasi-conforming finite element approximation for a fourth order variational inequality with displacement obstacle is considered, the optimal order of error estimate O(h) is obtained which is as same as that of the conventional finite elements.
基金Supported by National Natural Science Fund of China (11061021)Key Project of Chinese Ministry of Education (12024)+2 种基金Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108,2012MS0106,2011BS0102)Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011,NJZY13199)Program of Higher-level talents of Inner Mongolia University (125119,Z200901004,30105-125132)
文摘A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.
基金in part supported by the Distinguished Young Scholars Fund of Xinjiang Province(2013711010)NCET-13-0988the NSF of China(11271313,11271298,61163027,and 11362021)
文摘In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.
基金the National Natural Science Foundation of China Grants U1637208 and 71773024.the National Natural Science Foundation of China Grant 11971132.
文摘In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving waves.By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are shown.To derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are used.This,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+1.Numerical experiments are given to demonstrate the theoretical results.
基金Supported by the National Natural Science Foundation of China(11271340,116713697)Supported by Henan Natural Science Foundation of China(132300410376)
文摘In this paper, a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as approximation space for the velocity and the linear element for the pressure. The convergence analysis is presented and optimal error estimates of both broken H^1-norm and L^2-norm for velocity as well as the L^2-norm for the pressure are derived.
文摘In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.
文摘In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.
基金supported by National Natural Science Foundation of China(Grant Nos.11971132 and 11971131)Natural Science Foundation of Heilongjiang Province(Grant No.YQ2021A002)Guangdong Basic and Applied Basic Research Foundation(Grant No.2020B1515310006)。
文摘In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition,which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure.For the diffusive term,a pair of generalized alternating fluxes is considered.By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms,we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems.Numerical experiments including long time simulations,different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.
文摘Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.
基金supported by the National Natural Science Foundation of China(No.11671369,No.12071443)Key Scientific Research Project of Colleges and Universities in Henan Province(No.20B110013).
文摘The focus of this paper is on a linearized backward differential formula(BDF)scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations(KGSEs)with damping mechanism.Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme.The proof consists of three ingredients.First,a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms.Second,optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms.Third,by virtue of the relationship between the Ritz projection and the interpolation,as well as a so-called"lifting"technique,the superconvergence behavior of order O(h^(2)+τ^(2))in H^(1)-norm for the original variables are deduced.Finally,a numerical experiment is conducted to confirm our theoretical analysis.Here,h is the spatial subdivision parameter,andτis the time step.
基金supported by the National Natural Science Foundation of China under grants No.11971010,11771162,12231003.
文摘A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.