An hybrid finite volume-finite element method for variable density incompressible flows
Journal of Computational Physics
Notes on accuracy of finite-volume discretization schemes on irregular grids
Applied Numerical Mathematics
Remarks on the Consistency of Upwind Source at Interface Schemes on Nonuniform Grids
Journal of Scientific Computing
Error estimate for the upwind finite volume method for the nonlinear scalar conservation law
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
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This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes.