In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defin...In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.展开更多
In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from th...In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as H^(1) and L^(2) error estimates.The novelty of this paper is that,each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts:the first part is the element stiffness matrix of the standard quadratic finite element method(FEM),the second part is the difference between the FVE and FEM on the element boundary,while the third part can be expressed as the tensor product of two vectors.As a result,we reach a sufficient condition to guarantee the existence,uniqueness and coercivity result of the FVE solution on general triangular meshes.Moreover,based on this sufficient condition,some minimum angle conditions with simple,analytic and computable expressions are obtained.By comparison,the existing minimum angle conditions were obtained numerically from a computer program.Theoretical findings are conformed with the numerical results.展开更多
基金This work was partially supported by the National Natural Science Foundation of China(No.11871009).
文摘In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.
基金supported by the Guangdong Basic and Applied Basic Research Foundation,China(No.2022A1515012106)the project of Guangdong Polytechnic Normal University,China(No.2022SDKYA023)the project of promoting research capabilities for key constructed disciplines in Guangdong Province,China(No.2021ZDJS028).
文摘In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as H^(1) and L^(2) error estimates.The novelty of this paper is that,each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts:the first part is the element stiffness matrix of the standard quadratic finite element method(FEM),the second part is the difference between the FVE and FEM on the element boundary,while the third part can be expressed as the tensor product of two vectors.As a result,we reach a sufficient condition to guarantee the existence,uniqueness and coercivity result of the FVE solution on general triangular meshes.Moreover,based on this sufficient condition,some minimum angle conditions with simple,analytic and computable expressions are obtained.By comparison,the existing minimum angle conditions were obtained numerically from a computer program.Theoretical findings are conformed with the numerical results.
