Superconvergence of finite volume element method for elliptic problems. (English) Zbl 1298.65162
Summary: We study the superconvergence of finite volume element (FVE) method for elliptic problems by using linear trial functions. Under the condition of C-uniform meshes, we first establish a superclose weak estimate for the bilinear form of FVE method. Then, we prove that all interior mesh points are the optimal stress points of interpolation function and further we give the superconvergence result of gradient approximation: \(\max\limits_{P\in S}|(\nabla u-\overline\nabla u_h)(P)|=O\left(h^2\right)|\ln h|\), where \(S\) is the set of mesh points and \(\overline\nabla\) denotes the average gradient on elements containing vertex \(P\).
MSC:
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
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