A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element m...A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.展开更多
Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow pheno...Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.展开更多
This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite differenc...This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.展开更多
Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperatur...Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H1 norm error estimates are given.展开更多
In this paper, a biquartic finite volume element method based on Lobatto- Guass structure is presented for variable coefficient elliptic equation on rectangular partition. Not only the optimal H1 and L2 error estimate...In this paper, a biquartic finite volume element method based on Lobatto- Guass structure is presented for variable coefficient elliptic equation on rectangular partition. Not only the optimal H1 and L2 error estimates but also some superconvergent properties are available and could be proved for this method. The numerical results obtained by this finite volume element scheme confirm the validity of the theoretical analysis and the effectiveness of this method.展开更多
Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direc...Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.展开更多
In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method (FVE) based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.
In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under m...In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.展开更多
<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introdu...<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introducing intermediate variables, the original </span><span style="font-family:Verdana;">fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">.</span></span></span><span><span><span style="font-family:Verdana;"> Numerical examples are provided to confirm the theoretical results. In the end, we test the value of <em>δ</em></span><span style="font-family:Verdana;"> to observe its influence on the model.</span></span></span>展开更多
Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadrati...Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.展开更多
We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee ...We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(∥q∥_∞h+∥g∥_∞h~2)-acute and h is small enough,where h denotes the mesh size,q and g are coefficients of the convection and reaction terms,respectively.To deal with the convection-dominated problems,we use the upwind triangle technique.For such scheme,the mesh condition can be sharper to O(∥g∥_∞h~2)-acute.Some numerical examples are presented to demonstrate the theoretical results.展开更多
The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations,the pressure–velocity equation and the concentration equation.In this pa...The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations,the pressure–velocity equation and the concentration equation.In this paper,we present a mixed finite volume element method(FVEM)for the approximation of the pressure–velocity equation and a standard FVEM for the concentration equation.A priori error estimates in L^(∞)(L^(2))are derived for velocity,pressure and concentration.Numerical results are presented to substantiate the validity of the theoretical results.展开更多
In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained...In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained for the numerical solution in a discrete L^(2)-norm.A numerical experiment is presented to test the theoretical result.展开更多
A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term wit...A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.展开更多
In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic mu...In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.展开更多
A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer...A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced- order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illus- trate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.展开更多
In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the ...In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid.We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator.Finally,numerical simulations are provided to verify the correctness of the theoretical analysis.展开更多
文摘A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.
基金King Mongkut’s University of Technology North Bangkok (KMUTNB)the Office of the Higher Education Commission (OHEC)the National Metal and Materials Technology Center (MTEC) for supporting this research work
文摘Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.
文摘This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.
基金Supported by National Natural Science Foundation of China(10071044)the Research Fund of Doctoral Program of High Education by State Education Ministry of China.
文摘Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H1 norm error estimates are given.
文摘In this paper, a biquartic finite volume element method based on Lobatto- Guass structure is presented for variable coefficient elliptic equation on rectangular partition. Not only the optimal H1 and L2 error estimates but also some superconvergent properties are available and could be proved for this method. The numerical results obtained by this finite volume element scheme confirm the validity of the theoretical analysis and the effectiveness of this method.
基金supported by the National Natural Science Foundation of China grants No.11971241.
文摘Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.
基金The Major State Basic Research Program (19871051) of China and the NNSP (19972039) of China.
文摘In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method (FVE) based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.
基金The Major State Basic Research Program (19871051) of China the NNSF (19972039) of China and Yantai University Doctor Foundation (SX03B20).
文摘In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.
文摘<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introducing intermediate variables, the original </span><span style="font-family:Verdana;">fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">.</span></span></span><span><span><span style="font-family:Verdana;"> Numerical examples are provided to confirm the theoretical results. In the end, we test the value of <em>δ</em></span><span style="font-family:Verdana;"> to observe its influence on the model.</span></span></span>
基金supported by the National Natural Science Foundation of China(Nos.11871009,12271055)the Foundation of LCP and the Foundation of CAEP(CX20210044).
文摘Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.
基金the National Natural Science Foundation of China(No.11301033,11971069 and 12271209)the Natural Science Foundation of Jilin Province(No.20200201259JC)Jilin Province Science and Technology Plan Development Project(No.20210201078GX)。
文摘We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(∥q∥_∞h+∥g∥_∞h~2)-acute and h is small enough,where h denotes the mesh size,q and g are coefficients of the convection and reaction terms,respectively.To deal with the convection-dominated problems,we use the upwind triangle technique.For such scheme,the mesh condition can be sharper to O(∥g∥_∞h~2)-acute.Some numerical examples are presented to demonstrate the theoretical results.
文摘The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations,the pressure–velocity equation and the concentration equation.In this paper,we present a mixed finite volume element method(FVEM)for the approximation of the pressure–velocity equation and a standard FVEM for the concentration equation.A priori error estimates in L^(∞)(L^(2))are derived for velocity,pressure and concentration.Numerical results are presented to substantiate the validity of the theoretical results.
基金This work is supported by National Natural Science Foundation of China(Grant Nos.11271145 and 11031006)Foundation of Guizhou Science and Technology Department(Grant No.[2011]2098)+1 种基金Foundation for Talent Introduction of Guangdong Provincial University,Specialized Research Fund for the Doctoral Programof Higher Education(Grant No.20114407110009)the Project of Department of Education of Guangdong Province(Grant No.2012KJCX0036).
文摘In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained for the numerical solution in a discrete L^(2)-norm.A numerical experiment is presented to test the theoretical result.
基金supported by the Postdoctoral Science Foundation of China(No.2017M620689)the National Science Foundation of China(Nos.11571048,11401034)the CAEP developing fund of science and technology(No.2014A0202009).
文摘A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.
文摘In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.
基金supported by the National Science Foundation of China(12271227,11061009 and 11061021)Natural Science Foundation of Inner Mongolia(2012MS0106)+3 种基金Science Research Program of Guizhou(GJ[2011]2367)Science Research Program of Inner Mongolia Advanced Education(NJ10006)Special Funds for Co-construction Project of BeijingNorth China Electric Power University
文摘A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced- order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illus- trate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.
基金the helpful suggestions.This work was supported by National Natural Science Foundation of China(Nos.11771375,11971152)Shandong Province Natural Science Foundation(No.ZR2018MA008)NSF of Henan Province(No.202300410167).
文摘In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid.We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator.Finally,numerical simulations are provided to verify the correctness of the theoretical analysis.