This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A resid- ual error estimator is presented for the standard FE...This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A resid- ual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu es- timates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.展开更多
In this paper,we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model.The model consists of five nonlinear elliptic equations,and two of them describe q...In this paper,we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model.The model consists of five nonlinear elliptic equations,and two of them describe quantum corrections for quasi-Fermi levels.We propose an interpolated-exponential finite element(IEFE)method for solving the two quantum-correction equations.The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations.Moreover,we solve the two continuity equations with the edge-averaged finite element(EAFE)method to reduce numerical oscillations of quasi-Fermi levels.The Poisson equation of electrical potential is solved with standard Lagrangian finite elements.We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional.A Newton method is proposed to solve the nonlinear discrete problem.Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.展开更多
The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone...The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm,some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours.The multigrid V-cycle algorithm uses O(N)operations per iteration and is optimal.展开更多
We consider H(curl, Ω)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral...We consider H(curl, Ω)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 (Ω)-context along with local discrete Helmholtz-type decompositions of the edge element space.展开更多
In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-precondit...In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.展开更多
We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s equations.The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of fr...We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s equations.The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their“immediate”neighbors.In the context of lowest order conforming finite element approximations,we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms.The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures.The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom.We demonstrate our convergence theory by two numerical experiments.展开更多
In this paper,the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations.We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform conver...In this paper,the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations.We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes.A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations.It turns out to be robust for relatively large physical parameters.By extensive numerical experiments,we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.展开更多
In this paper,hp-adaptive finite element methods are studied for timeharmonic Maxwell’s equations.We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a...In this paper,hp-adaptive finite element methods are studied for timeharmonic Maxwell’s equations.We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates.Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities,edge singularities,and an engineering benchmark problem of Maxwell’s equations.The hp-adaptive methods show much better performance than the h-adaptive method.展开更多
Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions,where the wave propagation is governed by the Helmholtz equation.The scattering problem is modeled as a boundary value problem over a...Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions,where the wave propagation is governed by the Helmholtz equation.The scattering problem is modeled as a boundary value problem over a bounded domain.Based on the Dirichlet-to-Neumann(DtN)operator,a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle.An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition.Numerical experiments are included to compare with the perfectly matched layer(PML)method to illustrate the competitive behavior of the proposed adaptive method.展开更多
New-YearWorkshop on Scientific Computing 2019 is the second activity of the forum on scientific computing for young mathematicians,which was held in the last two days of 2018,in Xiangtan University in Chairman Mao’s ...New-YearWorkshop on Scientific Computing 2019 is the second activity of the forum on scientific computing for young mathematicians,which was held in the last two days of 2018,in Xiangtan University in Chairman Mao’s hometown-Xiangtan,Hunan Province,China.The first one was held just in the same dates of 2017.Xiangtan University was founded in 1958 according to Chairman Mao’s instruction.Though being away from any super-cities,the university attained amazingly an excellent repute in its academic records.Particularly,mathematics is one of its most outstanding areas all over China.A dozen of famous mathematicians,such as Yaxiang Yuan,Jinchao Xu,and Xiangyu Zhou,were graduated from Xiangtan University.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11471329,11321061,and 91430215)the National Magnetic Confinement Fusion Science Program of China(No.2015GB110000)+1 种基金the Youth Innovation Promotion Association of Chinese Academy of Sciences(CAS)(No.2016003)the National Center for Mathematics and Interdisciplinary Sciences of CAS
文摘This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A resid- ual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu es- timates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.
基金supported by National Key R&D Program of China 2019YFA0709600 and 2019YFA0709602Weiying Zheng was supported in part by National Key R&D Program of China 2019YFA0709600 and 2019YFA0709602the National Science Fund for Distinguished Young Scholars 11725106,and the NSFC major grant 11831016.
文摘In this paper,we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model.The model consists of five nonlinear elliptic equations,and two of them describe quantum corrections for quasi-Fermi levels.We propose an interpolated-exponential finite element(IEFE)method for solving the two quantum-correction equations.The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations.Moreover,we solve the two continuity equations with the edge-averaged finite element(EAFE)method to reduce numerical oscillations of quasi-Fermi levels.The Poisson equation of electrical potential is solved with standard Lagrangian finite elements.We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional.A Newton method is proposed to solve the nonlinear discrete problem.Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.
基金supported by the NSF of China (Grant Nos.12171238,12261160361)supported in part by the China NSF for Distinguished Young Scholars (Grant No.11725106)by the China NSF major project (Grant No.11831016).
文摘The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm,some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours.The multigrid V-cycle algorithm uses O(N)operations per iteration and is optimal.
基金supported in part by China NSF under the grant 60873177by the National Basic Research Project under the grant 2005CB321702
文摘We consider H(curl, Ω)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 (Ω)-context along with local discrete Helmholtz-type decompositions of the edge element space.
文摘In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.
基金supported in part by the National Magnetic Confinement Fusion Science Program(Grant No.2011GB105003)the NSF of China under the grants 91130004,11071116,and 10971096+1 种基金supported in part by China NSF under the grants 11031006 and 11171334the Funds for Creative Research Groups of China(Grant No.11021101).
文摘We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s equations.The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their“immediate”neighbors.In the context of lowest order conforming finite element approximations,we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms.The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures.The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom.We demonstrate our convergence theory by two numerical experiments.
基金supported in part by the National Science Fund for Dist inguished Young Scholars 11725106,China NSF grant 11831016.
文摘In this paper,the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations.We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes.A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations.It turns out to be robust for relatively large physical parameters.By extensive numerical experiments,we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.
基金supported in part by the National Basic Research Project under the grant 2011CB309703,by the Funds for Creative Research Groups of China(Grant No.11021101)by China NSF under the grant 60873177+2 种基金supported in part by China NSF under the grants 11031006 and 11171334by the Funds for Creative Research Groups of China(Grant No.11021101)by the National Magnetic Confinement Fusion Science Program(Grant No.2011GB105003).
文摘In this paper,hp-adaptive finite element methods are studied for timeharmonic Maxwell’s equations.We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates.Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities,edge singularities,and an engineering benchmark problem of Maxwell’s equations.The hp-adaptive methods show much better performance than the h-adaptive method.
基金supported in part by the NSF grants DMS-0914595 and DMS1042958supported in part by China NSF under the grants 11031006 and 11171334+1 种基金by the Funds for Creative Research Groups of China(Grant No.11021101)by the National Magnetic Confinement Fusion Science Program(Grant No.2011GB105003).
文摘Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions,where the wave propagation is governed by the Helmholtz equation.The scattering problem is modeled as a boundary value problem over a bounded domain.Based on the Dirichlet-to-Neumann(DtN)operator,a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle.An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition.Numerical experiments are included to compare with the perfectly matched layer(PML)method to illustrate the competitive behavior of the proposed adaptive method.
文摘New-YearWorkshop on Scientific Computing 2019 is the second activity of the forum on scientific computing for young mathematicians,which was held in the last two days of 2018,in Xiangtan University in Chairman Mao’s hometown-Xiangtan,Hunan Province,China.The first one was held just in the same dates of 2017.Xiangtan University was founded in 1958 according to Chairman Mao’s instruction.Though being away from any super-cities,the university attained amazingly an excellent repute in its academic records.Particularly,mathematics is one of its most outstanding areas all over China.A dozen of famous mathematicians,such as Yaxiang Yuan,Jinchao Xu,and Xiangyu Zhou,were graduated from Xiangtan University.