Considering the acoustic source scattering problems,when the sour-ce is non-radiating/invisible,we investigate the geometrical characterization for the underlying sources at polyhedral and conical corner.It is reveale...Considering the acoustic source scattering problems,when the sour-ce is non-radiating/invisible,we investigate the geometrical characterization for the underlying sources at polyhedral and conical corner.It is revealed that the non-radiating source with Holder continuous regularity must vanish at the corner.Using this kind of geometrical characterization of non-radiating sources,we establish local and global unique determination for a source with the polyhedral or corona shape support by a single far field measurement.Uniqueness by a single far field measurement constitutes of a long standing problem in inverse scattering problems.展开更多
The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We stud...The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.展开更多
This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a ...This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method.Recursive application of this idea results in the EXCMG method proposed in this paper.Finally,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.展开更多
基金the National Natural Science Foundation of China(Grant No.12371422)the Fundamental Research Funds for the Central Universities,JLU(Grant No.93Z172023Z01)+3 种基金The work of Y.Geng is supported by the Graduate Innovation Fund of Jilin University(Grant No.2023Cx276)the Hong Kong RGC General Research Funds(Projects 12302919,12301420 and 11300821)the NSFC/RGC Joint Research Fund(Project N.City U101/21)the France-Hong Kong ANR/RGCJoint Research(Grant No.A-HKBU 203/19).
文摘Considering the acoustic source scattering problems,when the sour-ce is non-radiating/invisible,we investigate the geometrical characterization for the underlying sources at polyhedral and conical corner.It is revealed that the non-radiating source with Holder continuous regularity must vanish at the corner.Using this kind of geometrical characterization of non-radiating sources,we establish local and global unique determination for a source with the polyhedral or corona shape support by a single far field measurement.Uniqueness by a single far field measurement constitutes of a long standing problem in inverse scattering problems.
基金supported in part by NSF grants DMS0604235 and DMS0906440the Research Fund of Indiana University.
文摘The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.
基金Kejia Pan was supported by the National Natural Science Foundation of China(Nos.41474103 and 41204082)the National High Technology Research and Development Program of China(No.2014AA06A602)+3 种基金the Natural Science Foundation of Hunan Province of China(No.2015JJ3148)Dongdong He was supported by the Fundamental Research Funds for the Central Universities,the National Natural Science Foundation of China(No.11402174)the Program for Young Excellent Talents at Tongji University(No.2013KJ012)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry。
文摘This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method.Recursive application of this idea results in the EXCMG method proposed in this paper.Finally,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.
