In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existen...In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.展开更多
In this paper, the convergence turbulent flow equations are considered. By rates of solutions to the three-dimensional combining the LP-Lq estimate for the linearized equations and an elaborate energy method, the conv...In this paper, the convergence turbulent flow equations are considered. By rates of solutions to the three-dimensional combining the LP-Lq estimate for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the equilibrium state is small in the H3-framework. More precisely, the optimal convergence rates of the solutions and their first-order derivatives in the L2-norm are obtained when the LP-norm of the perturbation is bounded for some p ε [1, 6).展开更多
The Cauchy problem of the Landau equation with frictional force is investigated. Based on Fourier analysis and nonlinear energy estimates, the optimal convergence rate to the steady state is obtained under some condit...The Cauchy problem of the Landau equation with frictional force is investigated. Based on Fourier analysis and nonlinear energy estimates, the optimal convergence rate to the steady state is obtained under some conditions on initial data.展开更多
In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic sys...In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.展开更多
The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear indepen...The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem,for a given multiplier vectorμ,the rate of convergence of the augmented Lagrangian method is linear with respect to||μu-μ^(*)||and the ratio constant is proportional to 1/c when the ratio|μ-μ^(*)||/c is small enough,where c is the penalty parameter that exceeds a threshold c_(*)>O andμ^(*)is the multiplier corresponding to a local minimizer.Moreover,we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters(ck)is bounded and the convergence rate is superlinear if(ck)is increasing to infinity.Finally,we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.展开更多
The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consis...The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L^2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1 + t)^-3/4 in L^2-norm, while the individual momentum of the charged particles converges at the optimal rate (1 + t)^-1/4 which is slower than the rate (1 + t)^-3/4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that almost counteracts the influence of the electric field so that the total density and momentum of the two carriers converges at a faster rate (1 + t)^-3/4+ε for any small constant ε 〉 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.展开更多
The isentropic bipolar compressible Navier-Stokes-Poisson (BNSP) system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data...The isentropic bipolar compressible Navier-Stokes-Poisson (BNSP) system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data belong to the Sobolev space H l(R3) ∩ B˙ s 1,1 (R3) with l ≥ 4 and s ∈ (0, 1], it is shown that the momenta of the charged particles decay at the optimal rate (1+t) 1 4 s 2 in L2 -norm, which is slower than the rate (1+t) 3 4 s 2 for the compressible Navier-Stokes (NS) equations [14]. In particular, a new phenomenon on the charge transport is observed. The time decay rate of total density and momentum was both (1 + t) 3 4 due to the cancellation effect from the interplay interaction of the charged particles.展开更多
This article concerded with a semiparametric generalized partial linear model (GPLM) with the type Ⅱ censored data. A sieve maximum likelihood estimator (MLE) is proposed to estimate the parameter component, allo...This article concerded with a semiparametric generalized partial linear model (GPLM) with the type Ⅱ censored data. A sieve maximum likelihood estimator (MLE) is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.展开更多
The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],...The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.展开更多
In this paper we propose a kind of implicit iterative methods for solving ill-posed operator equations and discuss the properties of the methods in the case that the control parameter is fixed. The theoretical results...In this paper we propose a kind of implicit iterative methods for solving ill-posed operator equations and discuss the properties of the methods in the case that the control parameter is fixed. The theoretical results show that the new methods have certain important features and can overcome some disadvantages of Tikhonov-type regularization and explicit iterative methods. Numerical examples are also given in the paper, which coincide well with theoretical results.展开更多
In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate var...In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate variableσ=∆u and translate the fourth-order derivative of the original problem into a second-order coupled system.We discretize the fractional time derivative terms by using the L1-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula.In the fully discrete scheme,we implement the finite element method for the spatial approximation.Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained.Numerical experiments are carried out to demonstrate our theoretical analysis.展开更多
We propose a method that combines isogeometric analysis(IGA)with the discontinuous Galerkin(DG)method for solving elliptic equations on 3-dimensional(3D)surfaces consisting of multiple patches.DG ideology is adopted a...We propose a method that combines isogeometric analysis(IGA)with the discontinuous Galerkin(DG)method for solving elliptic equations on 3-dimensional(3D)surfaces consisting of multiple patches.DG ideology is adopted across the patch interfaces to glue the multiple patches,while the traditional IGA,which is very suitable for solving partial differential equations(PDEs)on(3D)surfaces,is employed within each patch.Our method takes advantage of both IGA and the DG method.Firstly,the time-consuming steps in mesh generation process in traditional finite element analysis(FEA)are no longer necessary and refinements,including h-refinement and p-refinement which both maintain the original geometry,can be easily performed by knot insertion and order-elevation(Farin,in Curves and surfaces for CAGD,2002).Secondly,our method can easily handle the cases with non-conforming patches and different degrees across the patches.Moreover,due to the geometric flexibility of IGA basis functions,especially the use of multiple patches,we can get more accurate modeling of more complex surfaces.Thus,the geometrical error is significantly reduced and it is,in particular,eliminated for all conic sections.Finally,this method can be easily formulated and implemented.We generally describe the problem and then present our primal formulation.A new ideology,which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain(the former is easier to get),is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form,and the error analysis under both the DG norm and the L2 norm.The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L2 norm.Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.展开更多
A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical schem...A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.展开更多
To reduce the computational cost,we propose a regularizing modified LevenbergMarquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems.Convergence results for the regularizing modified L...To reduce the computational cost,we propose a regularizing modified LevenbergMarquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems.Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved.Based on these results,we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme.By imposing certain conditions on the noise,we derive optimal convergence rates on the approximate solution under special source conditions.Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.展开更多
In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discret...In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.展开更多
In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the l...In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.展开更多
基金supported by Strategic Research Grant of City University of Hong Kong, 7002129the Changjiang Scholar Program of Chinese Educational Ministry in Shanghai Jiao Tong University+1 种基金The research of the second author was supported partially by NSFC (10601018)partially by FANEDD
文摘In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.
基金supported by the National Natural Science Foundation of China(Nos.11071057 and 11271052)the Special Fund Project of Mathematical Tian Yuan Fund(No.11226029)
文摘In this paper, the convergence turbulent flow equations are considered. By rates of solutions to the three-dimensional combining the LP-Lq estimate for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the equilibrium state is small in the H3-framework. More precisely, the optimal convergence rates of the solutions and their first-order derivatives in the L2-norm are obtained when the LP-norm of the perturbation is bounded for some p ε [1, 6).
基金the first author is supported by the National Natural Science Foundation of China (11101188)the second author is supported by the National Natural Science Foundation of China (10871082)supported by the Fundamental Research Funds for the Central Universities
文摘The Cauchy problem of the Landau equation with frictional force is investigated. Based on Fourier analysis and nonlinear energy estimates, the optimal convergence rate to the steady state is obtained under some conditions on initial data.
基金Supported by National Natural Science Foundation of China-NSAF(10976026)the Research Funds for the Huaqiao Universities(12BS232)
文摘In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.
基金the National Natural Science Foundation of China(Nos.11991020,11631013,11971372,11991021,11971089 and 11731013)the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA27000000)Dalian High-Level Talent Innovation Project(No.2020RD09)。
文摘The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem,for a given multiplier vectorμ,the rate of convergence of the augmented Lagrangian method is linear with respect to||μu-μ^(*)||and the ratio constant is proportional to 1/c when the ratio|μ-μ^(*)||/c is small enough,where c is the penalty parameter that exceeds a threshold c_(*)>O andμ^(*)is the multiplier corresponding to a local minimizer.Moreover,we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters(ck)is bounded and the convergence rate is superlinear if(ck)is increasing to infinity.Finally,we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.
基金The research of the first author was partially supported by the NNSFC No.10871134the NCET support of the Ministry of Education of China+4 种基金the Huo Ying Dong Fund No.111033the Chuang Xin Ren Cai Project of Beijing Municipal Commission of Education #PHR201006107the Instituteof Mathematics and Interdisciplinary Science at CNUThe research of the second author was supported by the General Research Fund of Hong Kong (CityU 103109)the National Natural Science Foundation of China,10871082
文摘The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L^2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1 + t)^-3/4 in L^2-norm, while the individual momentum of the charged particles converges at the optimal rate (1 + t)^-1/4 which is slower than the rate (1 + t)^-3/4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that almost counteracts the influence of the electric field so that the total density and momentum of the two carriers converges at a faster rate (1 + t)^-3/4+ε for any small constant ε 〉 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.
基金supported by NSFC (10872004)National Basic Research Program of China (2010CB731500)the China Ministry of Education (200800010013)
文摘The isentropic bipolar compressible Navier-Stokes-Poisson (BNSP) system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data belong to the Sobolev space H l(R3) ∩ B˙ s 1,1 (R3) with l ≥ 4 and s ∈ (0, 1], it is shown that the momenta of the charged particles decay at the optimal rate (1+t) 1 4 s 2 in L2 -norm, which is slower than the rate (1+t) 3 4 s 2 for the compressible Navier-Stokes (NS) equations [14]. In particular, a new phenomenon on the charge transport is observed. The time decay rate of total density and momentum was both (1 + t) 3 4 due to the cancellation effect from the interplay interaction of the charged particles.
基金The talent research fund launched (3004-893325) of Dalian University of Technologythe NNSF (10271049) of China.
文摘This article concerded with a semiparametric generalized partial linear model (GPLM) with the type Ⅱ censored data. A sieve maximum likelihood estimator (MLE) is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.
基金The work of Yue is supported in part by NSF of China under the grants No.11971342.
文摘The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.
文摘In this paper we propose a kind of implicit iterative methods for solving ill-posed operator equations and discuss the properties of the methods in the case that the control parameter is fixed. The theoretical results show that the new methods have certain important features and can overcome some disadvantages of Tikhonov-type regularization and explicit iterative methods. Numerical examples are also given in the paper, which coincide well with theoretical results.
文摘In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate variableσ=∆u and translate the fourth-order derivative of the original problem into a second-order coupled system.We discretize the fractional time derivative terms by using the L1-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula.In the fully discrete scheme,we implement the finite element method for the spatial approximation.Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained.Numerical experiments are carried out to demonstrate our theoretical analysis.
基金Yan Xu:Research supported by NSFC grant No.11371342,No.11031007Fok Ying Tung Education Foundation No.131003+1 种基金Falai Chen:Research supported by NSFC grant No.11031007the National Basic Research Program of China(2011CB302400).
文摘We propose a method that combines isogeometric analysis(IGA)with the discontinuous Galerkin(DG)method for solving elliptic equations on 3-dimensional(3D)surfaces consisting of multiple patches.DG ideology is adopted across the patch interfaces to glue the multiple patches,while the traditional IGA,which is very suitable for solving partial differential equations(PDEs)on(3D)surfaces,is employed within each patch.Our method takes advantage of both IGA and the DG method.Firstly,the time-consuming steps in mesh generation process in traditional finite element analysis(FEA)are no longer necessary and refinements,including h-refinement and p-refinement which both maintain the original geometry,can be easily performed by knot insertion and order-elevation(Farin,in Curves and surfaces for CAGD,2002).Secondly,our method can easily handle the cases with non-conforming patches and different degrees across the patches.Moreover,due to the geometric flexibility of IGA basis functions,especially the use of multiple patches,we can get more accurate modeling of more complex surfaces.Thus,the geometrical error is significantly reduced and it is,in particular,eliminated for all conic sections.Finally,this method can be easily formulated and implemented.We generally describe the problem and then present our primal formulation.A new ideology,which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain(the former is easier to get),is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form,and the error analysis under both the DG norm and the L2 norm.The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L2 norm.Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.
文摘A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.
基金Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University(2020B1212060032)Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques(400440)+1 种基金the Foundation of Education Committee of Jiangxi,China(GJJ201436)National Natural Science Foundation of China under grants 11571386 and 11761010.
文摘To reduce the computational cost,we propose a regularizing modified LevenbergMarquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems.Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved.Based on these results,we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme.By imposing certain conditions on the noise,we derive optimal convergence rates on the approximate solution under special source conditions.Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.
基金supported in part by the Computational Physics Key Laboratory of IAPCAM(P.R.China)under Grant 6142A05200103(K.Cheng)the National Science Foundation(USA)under Grant NSF DMS-2012669(C.Wang)Grants NSF DMS-1719854,DMS-2012634(S.Wise).
文摘In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.
基金This work is supported in part by the grants NSFC 12071090(W.Chen)NSF DMS-2012669(C.Wang)+2 种基金NSFC 11871159Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001(X.Wang)NSF DMS-1719854,DMS-2012634(S.Wise).C.Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences,Fudan University,for the support.
文摘In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.