This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination ...This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.展开更多
The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element met...The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.展开更多
This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary condi...This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.展开更多
In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body...In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.展开更多
This paper is the third part in a series of papers on adaptive finite elementmethods based on optimal error estimates for linear elliptic problems on the concavecorner domains. In this paper, a result is obtained. The...This paper is the third part in a series of papers on adaptive finite elementmethods based on optimal error estimates for linear elliptic problems on the concavecorner domains. In this paper, a result is obtained. The algorithms for error controlboth in the energy norm and in the maximum norm presented in part 1 and part 2 ofthis series arc based on this result.展开更多
In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of fini...In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.展开更多
The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optima...The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.展开更多
An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any co...An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.展开更多
A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approxi...A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.展开更多
This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penaliz...This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.展开更多
The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully disc...The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.展开更多
A stabilized and convergent finite element formulation for the generalized Stokes problem is proposed and a posteriori analysis is performed to produce an error indicator. On this basis adaptive numerical method for s...A stabilized and convergent finite element formulation for the generalized Stokes problem is proposed and a posteriori analysis is performed to produce an error indicator. On this basis adaptive numerical method for solying the problem is developed . Numerical calculations are performed to confirm the reliability and effectiveness of the method.展开更多
In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a...In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λ- independent multiplicative constants; the Lame constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.展开更多
The adaptive element techniques of contact problem are studied by means of penalty method, and the error estimators are discussed. Based on error estimators, algorithm of the adaptive element techniques is developed, ...The adaptive element techniques of contact problem are studied by means of penalty method, and the error estimators are discussed. Based on error estimators, algorithm of the adaptive element techniques is developed, then the Gauss - Newton iterations are used which allow the nonlinear problem to be transformed into a sequence of linear sub- problems then easily solved. In addition, the algorithm can be applied into the simulation of de -bonding of fiber - reinforced composites.展开更多
A solution of probabilistic FEM for elastic-plastic materials is presented based on the incremental theory of plasticity and a modified initial stress method. The formulations are deduced through a direct differentiat...A solution of probabilistic FEM for elastic-plastic materials is presented based on the incremental theory of plasticity and a modified initial stress method. The formulations are deduced through a direct differentiation scheme. Partial differentiation of displacement, stress and the performance function can be iteratively performed with the computation of the mean values of displacement and stress. The presented method enjoys the efficiency of both the perturbation method and the finite difference method, but avoids the approximation during the partial differentiation calculation. In order to improve the efficiency, the adjoint vector method is introduced to calculate the differentiation of stress and displacement with respect to random variables. In addition, a time-saving computational method for reliability index of elastic-plastic materials is suggested based upon the advanced First Order Second Moment (FOSM) and by the usage of Taylor expansion for displacement. The suggested method is also applicable to 3-D cases.展开更多
The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.The perfectly matched layer(PML)technique is adopted to truncate the unbounded physical domain into a bounded comput...The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.The perfectly matched layer(PML)technique is adopted to truncate the unbounded physical domain into a bounded computational domain.An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem.Numerical experiments are included to demonstrate the efficiency of the proposed method.展开更多
A mixed finite element solution of contact stresses in meshing gears is investigated with the consideration of coupled thermo-elastic deformation and impact behavior. A simulation procedure of finite element solution ...A mixed finite element solution of contact stresses in meshing gears is investigated with the consideration of coupled thermo-elastic deformation and impact behavior. A simulation procedure of finite element solution of meshing gears is developed. The versatility of the procedure for both numerical accuracy and computational efficiency is verified by numerical analysis of meshing gear teeth.展开更多
This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based ...This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.展开更多
An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous solutions, conormal...An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous solutions, conormal derivatives, and coefficients. This algorithm iteratively solves small problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. Error estimates are derived to show that this algorithm always converges provided that relaxation parameters are suitably chosen. Numeric experiments with matching and non-matching grids at the interface from different phases are performed to show the accuracy of the method for capturing discontinuities in the solutions and coefficients. In contrast to standard numerical methods, the accuracy of our method does not seem to deteriorate as the coefficient discontinuity increases.展开更多
This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitt...This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.1120115911426102+4 种基金and 11571293)the Natural Science Foundation of Hunan Province(No.11JJ3135)the Foundation for Outstanding Young Teachers in Higher Education of Guangdong Province(No.Yq2013054)the Pearl River S&T Nova Program of Guangzhou(No.2013J2200063)the Construct Program of the Key Discipline in Hunan University of Science and Engineering
文摘This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.
文摘The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.
文摘This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.
基金supported by the US ARO grants 49308-MA and 56349-MAthe US AFSOR grant FA9550-06-1-024+1 种基金he US NSF grant DMS-0911434the State Key Laboratory of Scientific and Engineering Computing of Chinese Academy of Sciences during a visit by Z.Li between July-August,2008.
文摘In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.
文摘This paper is the third part in a series of papers on adaptive finite elementmethods based on optimal error estimates for linear elliptic problems on the concavecorner domains. In this paper, a result is obtained. The algorithms for error controlboth in the energy norm and in the maximum norm presented in part 1 and part 2 ofthis series arc based on this result.
基金Project supported by the National Natural Science Foundation of China(Nos.11671157 and11826212)
文摘In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.
基金Project supported by the National Natural Science Foundation of China(No.11271340)
文摘The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.
基金supported by the National Natural Science Foundation of China(Nos.10871156 and 11171269)the Fund of Xi'an Jiaotong University(No.2009xjtujc30)
文摘An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.
文摘A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.
文摘This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.
基金P.Sun was supported by NSF Grant DMS-1418806C.S.Zhang was partially supported by the National Key Research and Development Program of China(Grant No.2016YFB0201304)+1 种基金the Major Research Plan of National Natural Science Foundation of China(Grant Nos.91430215,91530323)the Key Research Program of Frontier Sciences of CAS.
文摘The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.
文摘A stabilized and convergent finite element formulation for the generalized Stokes problem is proposed and a posteriori analysis is performed to produce an error indicator. On this basis adaptive numerical method for solying the problem is developed . Numerical calculations are performed to confirm the reliability and effectiveness of the method.
文摘In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λ- independent multiplicative constants; the Lame constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.
文摘The adaptive element techniques of contact problem are studied by means of penalty method, and the error estimators are discussed. Based on error estimators, algorithm of the adaptive element techniques is developed, then the Gauss - Newton iterations are used which allow the nonlinear problem to be transformed into a sequence of linear sub- problems then easily solved. In addition, the algorithm can be applied into the simulation of de -bonding of fiber - reinforced composites.
基金The project supported by the Research Grant Council of Hong Kong (HKUST 722196E, 6039197E)the National Natural Science Foundation of China(59809003)the Foundation of University Key Teacher by the Chinese Ministry of Education
文摘A solution of probabilistic FEM for elastic-plastic materials is presented based on the incremental theory of plasticity and a modified initial stress method. The formulations are deduced through a direct differentiation scheme. Partial differentiation of displacement, stress and the performance function can be iteratively performed with the computation of the mean values of displacement and stress. The presented method enjoys the efficiency of both the perturbation method and the finite difference method, but avoids the approximation during the partial differentiation calculation. In order to improve the efficiency, the adjoint vector method is introduced to calculate the differentiation of stress and displacement with respect to random variables. In addition, a time-saving computational method for reliability index of elastic-plastic materials is suggested based upon the advanced First Order Second Moment (FOSM) and by the usage of Taylor expansion for displacement. The suggested method is also applicable to 3-D cases.
基金supported by China NSF grants 11771057,11401040,11671052supported by China NSF grants 1167105。
文摘The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.The perfectly matched layer(PML)technique is adopted to truncate the unbounded physical domain into a bounded computational domain.An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem.Numerical experiments are included to demonstrate the efficiency of the proposed method.
文摘A mixed finite element solution of contact stresses in meshing gears is investigated with the consideration of coupled thermo-elastic deformation and impact behavior. A simulation procedure of finite element solution of meshing gears is developed. The versatility of the procedure for both numerical accuracy and computational efficiency is verified by numerical analysis of meshing gear teeth.
基金supported by DOE grant DE-FE0009843National Natural Science Foundation of China(11175052)GRF of HKSAR#501012 and NSERC(Canada).
文摘This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.
文摘An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous solutions, conormal derivatives, and coefficients. This algorithm iteratively solves small problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. Error estimates are derived to show that this algorithm always converges provided that relaxation parameters are suitably chosen. Numeric experiments with matching and non-matching grids at the interface from different phases are performed to show the accuracy of the method for capturing discontinuities in the solutions and coefficients. In contrast to standard numerical methods, the accuracy of our method does not seem to deteriorate as the coefficient discontinuity increases.
文摘This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.
