The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element meth...The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.展开更多
In this paper,the electromagnetic scattering from overfilled cavities with inhomogeneous anisotropic media is investigated.To solve the scattering problem,a Petrov-Galerkin finite element interfacemethod on non-body-f...In this paper,the electromagnetic scattering from overfilled cavities with inhomogeneous anisotropic media is investigated.To solve the scattering problem,a Petrov-Galerkin finite element interfacemethod on non-body-fitted grids is presented.We reduce the infinite domain of scattering to a bounded domain problem by introducing a transparent boundary condition.The level set function is used to capture complex boundary and interface geometry that is not aligned with the mesh.Nonbody-fitted grids allow us to save computational costs during mesh generation and significantly reduce the amount of computer memory required.The solution is built by connecting two linear polynomials across the interfaces to satisfy the jump conditions.The proposed method can handle matrix coefficients produced by permittivity and permeability tensors of anisotropic media.The final linear system is sparse,making it more suitable for most iterative methods.Numerical experiments show that the proposed method has good convergence and realizability.Meanwhile,we discover that the absorbing properties of anisotropic media clearly and positively influence the reduction of radar cross section.It has also been demonstrated that the method can achieve second-order accuracy.展开更多
A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for bo...A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for both the fluid and the dispersed solid phases.Starting from the Navier-Stokes equations and a general description of the FEM strategy,the Streamline Upwind Petrov-Galerkin(SUPG)method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients.Then,the Variational Multiscale Method(VMS)is presented together with a detailed illustration of its algorithm and hierarchy of computational steps.It is demonstrated that the VMS can be considered as a more general version of the SUPG method.The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.展开更多
This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy in...This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy （semi）norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp （lower） upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.展开更多
A meshiess local discontinuous Petrov-Galerkin （MLDPG） method based on the local symmetric weak form （LSWF） is presented with the application to blasting problems. The derivation is similar to that of mesh-based R...A meshiess local discontinuous Petrov-Galerkin （MLDPG） method based on the local symmetric weak form （LSWF） is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin （RKDG） method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method （MM）, and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.展开更多
A numerical algorithm using a bilinear or linear finite element and semi-implicit three-step method is presented for the analysis of incompressible viscous fluid problems. The streamline upwind/Petrov-Galerkin (SUPG) ...A numerical algorithm using a bilinear or linear finite element and semi-implicit three-step method is presented for the analysis of incompressible viscous fluid problems. The streamline upwind/Petrov-Galerkin (SUPG) stabilization scheme is used for the formulation of the Navier-Stokes equations. For the spatial discretization, the convection term is treated explicitly, while the viscous term is treated implicitly, and for the temporal discretization, a three-step method is employed. The present method is applied to simulate the lid driven cavity problems with different geometries at low and high Reynolds numbers. The results compared with other numerical experiments are found to be feasible and satisfactory.展开更多
We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L2- and H1-norm that are explicit in the time steps,the ...We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L2- and H1-norm that are explicit in the time steps,the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover,we observe that the numerical scheme superconverges at the nodal points of the time partition.展开更多
In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the c...In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the cell are discussed.The condition number of the large sparse linear system is studied.Numerical results demonstrate that the method is nearly second order accurate in the L^(∞)norm and L^(2) norm,and is first order accurate in the H^(1) norm.展开更多
基金supported by the National Natural Science Foundation of China (No. 10871131)the Fund for Doctoral Authority of China (No. 200802700001)+1 种基金the Shanghai Leading Academic Discipline Project(No. S30405)the Fund for E-institutes of Shanghai Universities (No. E03004)
文摘The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.
基金supported by the National Natural Science Foundation of China(No.12271159)the Natural Science Foundation of Hebei Province(No.A2020502003)+2 种基金the Fundamental Research Funds for the Central Universities(No.2021MS115)supported by the National Natural Science Foundation of China(No.12171482)the State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum(No.PRP/DX-2307).
文摘In this paper,the electromagnetic scattering from overfilled cavities with inhomogeneous anisotropic media is investigated.To solve the scattering problem,a Petrov-Galerkin finite element interfacemethod on non-body-fitted grids is presented.We reduce the infinite domain of scattering to a bounded domain problem by introducing a transparent boundary condition.The level set function is used to capture complex boundary and interface geometry that is not aligned with the mesh.Nonbody-fitted grids allow us to save computational costs during mesh generation and significantly reduce the amount of computer memory required.The solution is built by connecting two linear polynomials across the interfaces to satisfy the jump conditions.The proposed method can handle matrix coefficients produced by permittivity and permeability tensors of anisotropic media.The final linear system is sparse,making it more suitable for most iterative methods.Numerical experiments show that the proposed method has good convergence and realizability.Meanwhile,we discover that the absorbing properties of anisotropic media clearly and positively influence the reduction of radar cross section.It has also been demonstrated that the method can achieve second-order accuracy.
基金The authors received the funding of the Royal Higher Institute for Defence(MSP16-06).
文摘A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for both the fluid and the dispersed solid phases.Starting from the Navier-Stokes equations and a general description of the FEM strategy,the Streamline Upwind Petrov-Galerkin(SUPG)method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients.Then,the Variational Multiscale Method(VMS)is presented together with a detailed illustration of its algorithm and hierarchy of computational steps.It is demonstrated that the VMS can be considered as a more general version of the SUPG method.The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.
基金Acknowledgments. The author was supported by the National Natural Science Foundation of China （11101013, 11401015） and the PHR （IHLB） under Grant PHR201108074.
文摘This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy （semi）norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp （lower） upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.
基金Supported by New Century Excellent Talents in University in China(NCET),National"973" Program(No.61338)Innovative Research Project of Xi'an Hi-Tech Institute(EPXY0806)
文摘A meshiess local discontinuous Petrov-Galerkin （MLDPG） method based on the local symmetric weak form （LSWF） is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin （RKDG） method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method （MM）, and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.
基金Project supported by the National Natural Science Foundation of China (No.51078230)the Research Fund for the Doctoral Program of Higher Education of China (No.200802480056)the Key Project of Fund of Science and Technology Development of Shanghai (No.10JC1407900),China
文摘A numerical algorithm using a bilinear or linear finite element and semi-implicit three-step method is presented for the analysis of incompressible viscous fluid problems. The streamline upwind/Petrov-Galerkin (SUPG) stabilization scheme is used for the formulation of the Navier-Stokes equations. For the spatial discretization, the convection term is treated explicitly, while the viscous term is treated implicitly, and for the temporal discretization, a three-step method is employed. The present method is applied to simulate the lid driven cavity problems with different geometries at low and high Reynolds numbers. The results compared with other numerical experiments are found to be feasible and satisfactory.
基金supported by National Natural Science Foundation of China(Grant Nos.11226330 and 11301343)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20113127120002)+3 种基金the Research Fund for Young Teachers Program in Shanghai(GrantNo.shsf018)the Fund for E-institute of Shanghai Universities(Grant No.E03004)supported by the Natural Sciences and Engineering Research Council of Canada(Grant No.OGP0046726)Shanghai University under Leading Academic Discipline Project of Shanghai MunicipalEducation Commission(Grant No.J50101)
文摘We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L2- and H1-norm that are explicit in the time steps,the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover,we observe that the numerical scheme superconverges at the nodal points of the time partition.
基金The author would like to thank the referees for the helpful suggestions.L.Shi’s research is supported by National Natural Science Foundation of China(No.11701569)L.Wang’s research is supported by Science Foundation of China University of Petroleum-Beijing(No.2462015BJB05).S.Hou’s research is supported by Dr.Walter Koss Endowed Professorship.This professorship is made available through the State of Louisiana Board of Regents Support Funds.
文摘In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the cell are discussed.The condition number of the large sparse linear system is studied.Numerical results demonstrate that the method is nearly second order accurate in the L^(∞)norm and L^(2) norm,and is first order accurate in the H^(1) norm.
