Resonant tunneling diodes(RTDs)exhibit a distinctive characteristic known as negative resistance.Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices.This paper conducts ...Resonant tunneling diodes(RTDs)exhibit a distinctive characteristic known as negative resistance.Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices.This paper conducts a thorough investigation into the currentvoltage(Ⅰ-Ⅴ)characteristics of RTDs utilizing various numerical methods.Through a series of numerical experiments,we verified that the transfer matrix method ensures robust convergence in Ⅰ-Ⅴcurves and proficiently determines the tunneling bias for energy potential functions with discontinuities.Our numerical analysis underscores the significant impact of variations in effective mass on Ⅰ-Ⅴ curves,emphasizing the need to consider this effect.Furthermore,we observe that increasing the doping concentration results in a reduction in tunneling bias and an enhancement in peak current.Leveraging the unique features of the Ⅰ-Ⅴ curve,we employ shallow neural networks to accurately fit the Ⅰ-Ⅴ curves,yielding satisfactory results with limited data.展开更多
In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the ...In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.展开更多
Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the re...Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.展开更多
Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v...Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the tem- perature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.展开更多
We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one. Furthermore, we apply this result to the theory of extremal quasiconformal mappings. Let [μ] be a point in the universal Teichmiille...We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one. Furthermore, we apply this result to the theory of extremal quasiconformal mappings. Let [μ] be a point in the universal Teichmiiller space such that the Hausdorff dimension of fμ（δ△） is bigger than one. We show that for every kn ∈ （0, 1） and polygonal differentials δn, n = 1, 2, the sequence {[kn δn/｜δn｜} cannot converge to [μ] under the Teichmiiller metric.展开更多
The lowest degree of polynomial for a finite element to solve a 2^th-order elliptic equation is k.The Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic equation.We de...The lowest degree of polynomial for a finite element to solve a 2^th-order elliptic equation is k.The Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic equation.We design a cubic H3-nonconforming macro-element on two-dimensional triangular grids,solving a sixth-order tri-harmonic equation.We also write down explicitly the 12 basis functions on each macro-element.A convergence theory is established and verified by numerical tests.展开更多
This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspa...This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.展开更多
In this paper,we introduce an operator Hμ(z)on L^∞(△)and obtain some of its properties.Some applications of this operator to the extremal problem of quasiconformal mappings are given.In particular,a sufficient cond...In this paper,we introduce an operator Hμ(z)on L^∞(△)and obtain some of its properties.Some applications of this operator to the extremal problem of quasiconformal mappings are given.In particular,a sufficient condition for a point r in the universal Teichmfiller space T(△)to be a Strebel point is obtained.展开更多
In this paper,we try to describe the relationship between the differentiability of a quasisymmetric homeomorphism and the local Hausdorff dimension of the quasiline at a point.
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divd...This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis.展开更多
The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unb...The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity.This hinders the interpretation of the results obtained by minimizing the OM functional.We provide a new perspective on this issue.Under mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of curves.The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal energy.We further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path simultaneously.This algorithm is successfully applied to several typical examples in rare event studies.Some interesting comparisons with the Freidlin-Wentzell action functional are also made.展开更多
In this paper,we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem.We derive robust a post...In this paper,we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem.We derive robust a posteriori error estimators for them and prove their reliability and efficiency.展开更多
The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem,whose optimal value is referred to as the dual Cheeger constant h^(+),and later i...The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem,whose optimal value is referred to as the dual Cheeger constant h^(+),and later improved through its modification h^(+).However,the dual Cheeger problem and its modification themselves are relatively unexplored,especially the lack of effective approximate algorithms.To this end,we first derive equivalent spectral formulations of h^(+)and h^(+)within the framework of the nonlinear spectral theory of signless 1-Laplacian,present their interactions with the Laplacian matrix and 1-Laplacians,and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved.We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector,and provide the approximate values of h^(+)and h^(+)on the G-set for the first time.The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating it into the inverse power algorithms,leading to significantly improved approximate values on the G-set.Finally,we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework cannot be improved beyond 0.769.展开更多
In this paper,we propose mixed finite element methods for the Reissner-Mindlin Plate Problem by introducing the bending moment as an independent variable.We apply the finite element approximations of the stress field ...In this paper,we propose mixed finite element methods for the Reissner-Mindlin Plate Problem by introducing the bending moment as an independent variable.We apply the finite element approximations of the stress field and the displacement field constructed for the elasticity problem by Hu(J Comp Math 33:283–296,2015),Hu and Zhang(arXiv:1406.7457,2014)to solve the bending moment and the rotation for the Reissner-Mindlin Plate Problem.We propose two triples of finite element spaces to approximate the bending moment,the rotation,and the displacement.The feature of these methods is that they need neither reduction terms nor penalty terms.Then,we prove the well-posedness of the discrete problem and obtain the optimal estimates independent of the plate thickness.Finally,we present some numerical examples to demonstrate the theoretical results.展开更多
A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated b...A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.展开更多
This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian.In virtue of the cell structure of the feasible set,we propose a cell descend(CD)framework for ach...This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian.In virtue of the cell structure of the feasible set,we propose a cell descend(CD)framework for achieving the Cheeger cut.While plugging the relaxation to guarantee the decrease of the objective value in the feasible set,from which both the inverse power(IP)method and the steepest descent(SD)method can also be recovered,we are able to get two specified CD methods.Comparisons of all these methods are conducted on several typical graphs.展开更多
A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes w...A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.展开更多
This paper is devoted to a new error analysis of nonconforming finite element methods.Compared with the classic error analysis in literature,only weak continuity,the F-E-M-Test for nonconforming finite element spaces,...This paper is devoted to a new error analysis of nonconforming finite element methods.Compared with the classic error analysis in literature,only weak continuity,the F-E-M-Test for nonconforming finite element spaces,and basic Hm regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed.The analysis is motivated by ideas from a posteriori error estimates and projection average operators.One main ingredient is a novel decomposition for some key average terms on(n.1)-dimensional faces by introducing a piecewise constant projection,which defines the generalization to more general nonconforming finite elements of the results in literature.The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.展开更多
The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blonders technique. We give a complete answer to the open problem proposed by Patel ...The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blonders technique. We give a complete answer to the open problem proposed by Patel et al., which automatically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature.展开更多
In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynom...In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynomials for all l≥3 and the corresponding finite element solution converges at the order l-1 in H2 norm.Moreover,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,respectively.The error estimate is proved.The numerical results are.provided to confirm the theoretical findings.展开更多
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171035,12371389,12471378)by the Natural Science Foundation of Guangdong Province of China(Grant No.2024A1515010356).
文摘Resonant tunneling diodes(RTDs)exhibit a distinctive characteristic known as negative resistance.Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices.This paper conducts a thorough investigation into the currentvoltage(Ⅰ-Ⅴ)characteristics of RTDs utilizing various numerical methods.Through a series of numerical experiments,we verified that the transfer matrix method ensures robust convergence in Ⅰ-Ⅴcurves and proficiently determines the tunneling bias for energy potential functions with discontinuities.Our numerical analysis underscores the significant impact of variations in effective mass on Ⅰ-Ⅴ curves,emphasizing the need to consider this effect.Furthermore,we observe that increasing the doping concentration results in a reduction in tunneling bias and an enhancement in peak current.Leveraging the unique features of the Ⅰ-Ⅴ curve,we employ shallow neural networks to accurately fit the Ⅰ-Ⅴ curves,yielding satisfactory results with limited data.
基金supported by National Natural Science Foundation of China (GrantNo.10971005)A Foundation for the Author of National Excellent Doctoral Dissertation of PR China (GrantNo.200718)+1 种基金supported in part by National Natural Science Foundation of China Key Project (Grant No.11031006)the Chinesisch-Deutsches Zentrum Project (Grant No.GZ578)
文摘In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.
文摘Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.
基金supported by National Natural Science Foundation of China(Grant Nos.50930003 and 21274005)
文摘Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the tem- perature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.
基金supported by National Natural Science Foundation of China(Grant Nos.10831004 and 11171080)
文摘We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one. Furthermore, we apply this result to the theory of extremal quasiconformal mappings. Let [μ] be a point in the universal Teichmiiller space such that the Hausdorff dimension of fμ（δ△） is bigger than one. We show that for every kn ∈ （0, 1） and polygonal differentials δn, n = 1, 2, the sequence {[kn δn/｜δn｜} cannot converge to [μ] under the Teichmiiller metric.
基金the National Natural Science Foundation of China(Nos.11271035,91430213,11421101).
文摘The lowest degree of polynomial for a finite element to solve a 2^th-order elliptic equation is k.The Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic equation.We design a cubic H3-nonconforming macro-element on two-dimensional triangular grids,solving a sixth-order tri-harmonic equation.We also write down explicitly the 12 basis functions on each macro-element.A convergence theory is established and verified by numerical tests.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371038, 11471025, 11421101 and 61121002)
文摘This is primarily an expository paper surveying up-to-date known results on the spectral theory of1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.
基金Supported by the National Science Foundation of China(Grants No.10171003 and 10231040)the Doctoral Education Program Foundation of China
文摘In this paper,we introduce an operator Hμ(z)on L^∞(△)and obtain some of its properties.Some applications of this operator to the extremal problem of quasiconformal mappings are given.In particular,a sufficient condition for a point r in the universal Teichmfiller space T(△)to be a Strebel point is obtained.
基金supported by National Natural Science Foundation of China(Grant Nos.11401432 and11571172)the second author is supported by National Natural Science Foundation of China(Grant No.11371035)
文摘In this paper,we try to describe the relationship between the differentiability of a quasisymmetric homeomorphism and the local Hausdorff dimension of the quasiline at a point.
基金supported by National Natural Science Foundation of China(Grant Nos.11625101 and 11421101).
文摘This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis.
基金supported by National Natural Science Foundation of China(Grant Nos.11421101,91530322 and 11825102)supported by the Construct Program of the Key Discipline in Hunan Province+1 种基金supported by Singapore Ministry of Education Academic Research Funds(Grant Nos.R-146-000-267-114 and R-146-000-232-112)National Natural Science Foundation of China(Grant No.11871365)。
文摘The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity.This hinders the interpretation of the results obtained by minimizing the OM functional.We provide a new perspective on this issue.Under mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of curves.The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal energy.We further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path simultaneously.This algorithm is successfully applied to several typical examples in rare event studies.Some interesting comparisons with the Freidlin-Wentzell action functional are also made.
基金J.Hu was supported by the NSFC project 10971005A Foundation for the Author of National Excellent Doctoral Dissertation of PR China 200718+1 种基金Y.Q.Huang was supported in part by the NSFC Key Project 11031006Hunan Provincial NSF Project 10JJ7001.
文摘In this paper,we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem.We derive robust a posteriori error estimators for them and prove their reliability and efficiency.
基金supported by the National Key R&D Program of China(No.2022YFA 1005102)National Natural Science Foundation of China(Grant Nos.12401443,12325112 and 12288101).
文摘The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem,whose optimal value is referred to as the dual Cheeger constant h^(+),and later improved through its modification h^(+).However,the dual Cheeger problem and its modification themselves are relatively unexplored,especially the lack of effective approximate algorithms.To this end,we first derive equivalent spectral formulations of h^(+)and h^(+)within the framework of the nonlinear spectral theory of signless 1-Laplacian,present their interactions with the Laplacian matrix and 1-Laplacians,and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved.We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector,and provide the approximate values of h^(+)and h^(+)on the G-set for the first time.The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating it into the inverse power algorithms,leading to significantly improved approximate values on the G-set.Finally,we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework cannot be improved beyond 0.769.
文摘In this paper,we propose mixed finite element methods for the Reissner-Mindlin Plate Problem by introducing the bending moment as an independent variable.We apply the finite element approximations of the stress field and the displacement field constructed for the elasticity problem by Hu(J Comp Math 33:283–296,2015),Hu and Zhang(arXiv:1406.7457,2014)to solve the bending moment and the rotation for the Reissner-Mindlin Plate Problem.We propose two triples of finite element spaces to approximate the bending moment,the rotation,and the displacement.The feature of these methods is that they need neither reduction terms nor penalty terms.Then,we prove the well-posedness of the discrete problem and obtain the optimal estimates independent of the plate thickness.Finally,we present some numerical examples to demonstrate the theoretical results.
基金supported by National Natural Science Foundation of China(Grant Nos.11271035,91430213 and 11421101)
文摘A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.
文摘This paper presents a detailed review of both theory and algorithms for the Cheeger cut based on the graph 1-Laplacian.In virtue of the cell structure of the feasible set,we propose a cell descend(CD)framework for achieving the Cheeger cut.While plugging the relaxation to guarantee the decrease of the objective value in the feasible set,from which both the inverse power(IP)method and the steepest descent(SD)method can also be recovered,we are able to get two specified CD methods.Comparisons of all these methods are conducted on several typical graphs.
基金supported by DFG Research Center MATHEON"Mathematics for key technologies" in Berlinsupported by the NSFC under Grant 10601003 and A Foundation for the Author of National Excellent Doctoral Dissertation of PR China 200718support of two Sino-German workshops on Applied and Computational Mathematics held in 2005 and 2007 through the Sino-German office in Beijing.
文摘A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.
基金supported by National Natural Science Foundation of China(Grant Nos.11031006 and 11271035)
文摘This paper is devoted to a new error analysis of nonconforming finite element methods.Compared with the classic error analysis in literature,only weak continuity,the F-E-M-Test for nonconforming finite element spaces,and basic Hm regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed.The analysis is motivated by ideas from a posteriori error estimates and projection average operators.One main ingredient is a novel decomposition for some key average terms on(n.1)-dimensional faces by introducing a piecewise constant projection,which defines the generalization to more general nonconforming finite elements of the results in literature.The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 60572056, 60528007, 60334020, 60204006, 10471044, and 10372002)the National Key Basic Research and Development Program (Grant Nos. 2005CB321902, 2004CB318003, 2002CB312200)+1 种基金the Overseas Outstanding Young Researcher Foundation of Chinese Academy of Sciencesthe Program of National Key Laboratory of Intelligent Technology and Systems of Tsinghua University
文摘The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blonders technique. We give a complete answer to the open problem proposed by Patel et al., which automatically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature.
基金supported by National Natural Science Foundation of China(Grant Nos.11625101 and 11421101)。
文摘In this article,a family of H^2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3 D.In the family,the Pl polynomial space is enriched by some high order polynomials for all l≥3 and the corresponding finite element solution converges at the order l-1 in H2 norm.Moreover,the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces,respectively.The error estimate is proved.The numerical results are.provided to confirm the theoretical findings.
