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On the accuracy of the finite volume element method based on piecewise linear polynomials. (English) Zbl 1036.65084
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 $H^1$ and $L^2$ error estimates and superconvergence are also discussed. Some examples are given to show that FVE method cannot have the standard $O(h^2)$ convergence rate in the $L^2$ norm when the source term has the minimum regularity in $L^2$, even if the exact solution is in $H^2$. The interested reader could also refer to {\it R. E. Ewing}, {\it Z. Li}, {\it T. Lin} and {\it Y. Lin} [Math. Comput. Simul. 50 , 63--76 (1999; Zbl 1027.65155)] and {\it T. Kerkhoven} [SIAM J. Numer. Anal. 33, 1864--1884 (1996; Zbl 0860.65101)].

65N15Error bounds (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
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