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A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids.(English)Zbl 1086.65108

The authors consider the finite volume method to approximately solve the two-dimensional Laplace equation \(-\Delta \phi =f\) on a bounded domain \(\Omega\) using two different meshes. Then they define on both meshes discrete and gradient divergence operators verifying the discrete Green formula. One of the main results of the paper is to show that this finite volume method is equivalent to a nonconforming finite element method. This allows the authors to prove first order convergence in both \(H_{0} ^{1}\) and \(L^{2}\) norms. On special grids they are even able to prove superconvergence of order two. Numerical experiments are carried out. The work concludes by presenting convergence curves in the discrete \(H_{0} ^{1}\) and \(L^{2}\) norms on various types of grids of triangular type or nonconforming.

MSC:

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
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

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References:

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[26]Special issue on the simulation of transport around a nuclear waste disposal site: the Couplex test cases. Computat. Geosci. 8 (2004). Zbl 1062.86501 ·Zbl 1062.86501 ·doi:10.1023/B:COMG.0000035097.89798.f9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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