Is \(2k\)-conjecture valid for finite volume methods? (English) Zbl 1327.65216
Summary: This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove the \(2k\)-conjecture: at each vertex of the underlying rectangular mesh, the bi-\(k\) degree finite volume solution approximates the exact solution with an order \( O(h^{2k})\), where \(h\) is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments.
MSC:
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
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