We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary cond...We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary condition for a k-order FVE scheme to have the optimal L^(2) convergence rate in 1D,in which the independent dual parameters constitute a(k-1)-dimension surface in the reasonable domain in k-dimension.In the analysis,the dual strategies in different primary elements are not necessarily to be the same,and they are allowed to be asymmetric in each primary element,which open up more possibilities of the FVE schemes to be applied to some complex problems,such as the convection-dominated problems.It worth mentioning that,the construction can be extended to the quadrilateral meshes in 2D.The stability and H^(1) estimate are proved for completeness.All the above results are demon-strated by numerical experiments.展开更多
The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not s...The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.展开更多
This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we o...This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we obtain an L^(P)-result(p>2),including the existence and uniqueness of the p-th power integrable solution,a p-th power estimate,and a related continuous dependence property of the solution on the coefficients,for coupled linear FBSDEs in the monotonicity framework over large time intervals.In order to get rid of the stubborn constraint commonly existing in the literature,i.e.,the Lipschitz constant of σ with respect to z is very small,we introduce a linear transformation to overcome the difficulty on small intervals,and then"splice"the L^(P)-results obtained on many small intervals to yield the desired one on a large interval.展开更多
In this paper, the authors give the local L^2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equa...In this paper, the authors give the local L^2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R^+→R satisfies some suitable conditions and φ((-?)^(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L^2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.展开更多
The authors establish weighted L^2-estimates of solutions for the damped wave equations with variable coefficients utt-div A(x)▽u + au_t = 0 in IR^nunder the assumption a(x) ≥ a_0[1 + ρ(x)]^(-l),where a_0 > 0, l...The authors establish weighted L^2-estimates of solutions for the damped wave equations with variable coefficients utt-div A(x)▽u + au_t = 0 in IR^nunder the assumption a(x) ≥ a_0[1 + ρ(x)]^(-l),where a_0 > 0, l < 1, ρ(x) is the distance function of the metric g = A^(-1)(x) on IR^n. The authors show that these weighted L^2-estimates are closely related to the geometrical properties of the metric g = A^(-1)(x).展开更多
In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We ...In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2) error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11431013,11825101,11522101,11688101)the National Key R&D Program of China(No.2021YFA1003100)。
文摘In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.
基金This work was supported in part by the NSFC under grant No.11701211.
文摘We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary condition for a k-order FVE scheme to have the optimal L^(2) convergence rate in 1D,in which the independent dual parameters constitute a(k-1)-dimension surface in the reasonable domain in k-dimension.In the analysis,the dual strategies in different primary elements are not necessarily to be the same,and they are allowed to be asymmetric in each primary element,which open up more possibilities of the FVE schemes to be applied to some complex problems,such as the convection-dominated problems.It worth mentioning that,the construction can be extended to the quadrilateral meshes in 2D.The stability and H^(1) estimate are proved for completeness.All the above results are demon-strated by numerical experiments.
基金supported by the National Key R&D Program of China(No.2021YFA1003001).
文摘The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871310,12271304 and 11971262)the Natural Science Foundation of Shandong Province(Grant No.ZR2020MA014)。
文摘This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we obtain an L^(P)-result(p>2),including the existence and uniqueness of the p-th power integrable solution,a p-th power estimate,and a related continuous dependence property of the solution on the coefficients,for coupled linear FBSDEs in the monotonicity framework over large time intervals.In order to get rid of the stubborn constraint commonly existing in the literature,i.e.,the Lipschitz constant of σ with respect to z is very small,we introduce a linear transformation to overcome the difficulty on small intervals,and then"splice"the L^(P)-results obtained on many small intervals to yield the desired one on a large interval.
基金supported by the National Natural Science Foundation of China(Nos.11571160,11661061,11761054)the Inner Mongolia University Scientific Research Projects(Nos.NJZZ16234,NJZY17289)
文摘In this paper, the authors give the local L^2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R^+→R satisfies some suitable conditions and φ((-?)^(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L^2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.
基金supported by the National Science Foundation of China under Grant Nos.61573342,61473126the Key Research Program of Frontier Sciences,Chinese Academy of Sciences,under Grant No.QYZDJ-SSWSYS011 the Fundamental Research Funds for the Central Universities
文摘The authors establish weighted L^2-estimates of solutions for the damped wave equations with variable coefficients utt-div A(x)▽u + au_t = 0 in IR^nunder the assumption a(x) ≥ a_0[1 + ρ(x)]^(-l),where a_0 > 0, l < 1, ρ(x) is the distance function of the metric g = A^(-1)(x) on IR^n. The authors show that these weighted L^2-estimates are closely related to the geometrical properties of the metric g = A^(-1)(x).
文摘In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2) error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.
