A symmetric finite volume scheme for selfadjoint elliptic problems

Based on a linear finite element space, a symmetric finite volume scheme for a self-adjoint elliptic boundary-value problem is proposed. Error estimates in L2-norm, H1-norm, and L∞-norm are derived. Some post-processing techniques are also provided.

[1]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[2]  B. Heinrich.  Finite Difference Methods on Irregular Networks , 1987 .

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

[4]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[5]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis , 1991 .

[6]  Zhiqiang Cai,et al.  The finite volume element method for diffusion equations on general triangulations , 1991 .

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

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

[9]  Aihui Zhou,et al.  The full approximation accuracy for the stream function-vorticity-pressure method , 1994 .

[10]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids , 1991 .

[11]  Ningning Yan,et al.  Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes , 2001 .

[12]  Bo Li,et al.  Analysis of a Class of Superconvergence Patch Recovery Techniques for Linear and Bilinear Finite Elements , 1999 .

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

[14]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

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

[16]  Pekka Neittaanmäki,et al.  On superconvergence techniques , 1987 .

[17]  Gabriel Wittum,et al.  Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes , 2001, Math. Comput..

[18]  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..

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