On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

[1]  A. A. Samarskii,et al.  Homogeneous difference schemes , 1962 .

[2]  A. A. Samarskii,et al.  Homogeneous difference schemes on non-uniform nets☆ , 1963 .

[3]  B. R. Baliga,et al.  A NEW FINITE-ELEMENT FORMULATION FOR CONVECTION-DIFFUSION PROBLEMS , 1980 .

[4]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[5]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[6]  Zhiqiang Cai,et al.  On the finite volume element method , 1990 .

[7]  Zhiqiang Cai,et al.  On the accuracy of the finite volume element method for diffusion equations on composite grids , 1990 .

[8]  Endre Süli.  Convergence of finite volume schemes for Poisson's equation on nonuniform meshes , 1991 .

[9]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .

[10]  Thomas Kerkhoven,et al.  Piecewise Linear Petrov--Galerkin Error Estimates For The Box Method , 1996 .

[11]  Junping Wang,et al.  Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions , 1996, Math. Comput..

[12]  Ilya D. Mishev,et al.  FINITE VOLUME METHODS ON VORONOI MESHES , 1998 .

[13]  Qiding Zhu.  A survey of superconvergence techniques in finite element methods , 1998 .

[14]  Rolf Stenberg,et al.  Finite element methods: superconvergence, post-processing, and a posteriori estimates , 1998 .

[15]  Huang Jianguo,et al.  On the Finite Volume Element Method for General Self-Adjoint Elliptic Problems , 1998 .

[16]  Chuanmiao Chen.  Element analysis method and superconvergence , 1998 .

[17]  Zhilin Li,et al.  The immersed finite volume element methods for the elliptic interface problems , 1999 .

[18]  Panagiotis Chatzipantelidis,et al.  A finite volume method based on the Crouzeix–Raviart element for elliptic PDE's in two dimensions , 1999, Numerische Mathematik.

[19]  Richard E. Ewing,et al.  Element Approximations of Nonlocal in Time One � dimensional Flows in Porous Media , 1998 .

[20]  Ronghua Li.  Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods , 2000 .

[21]  Qian Li,et al.  Error estimates in L2, H1 and Linfinity in covolume methods for elliptic and parabolic problems: A unified approach , 1999, Math. Comput..

[22]  James H. Bramble,et al.  The analysis of multigrid methods , 2000 .

[23]  R. Lazarov,et al.  Finite volume element approximations of nonlocal reactive flows in porous media , 2000 .

[24]  Richard E. Ewing,et al.  Mortar Finite Volume Element Methods And Domain Decompositions , 2000 .