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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 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)].

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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