The accuracy of the finite volume element (FVE) methods for solving second-order elliptic boundary value problems is studied. The approach presented herein combines traditional finite element and finite difference methods as a variation of the Galerkin finite element method, revealing regularities in the exact solution and establishing that the source term can affect the accuracy of FVE methods.
Optimal order and error estimates and superconvergence are also discussed. Some examples are given to show that FVE method cannot have the standard convergence rate in the norm when the source term has the minimum regularity in , even if the exact solution is in .
The interested reader could also refer to R. E. Ewing, Z. Li, T. Lin and Y. Lin [Math. Comput. Simul. 50 , 63–76 (1999; Zbl 1027.65155)] and T. Kerkhoven [SIAM J. Numer. Anal. 33, 1864–1884 (1996; Zbl 0860.65101)].