Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions

A Kinfack Jeutsa,J Nganhou,A Njifenjou

doi:10.1155/2016/5891064

Abstract

A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms andL2-norm), are investigated. Numerical test is provided.

Highlights

Introduction and the Model ProblemEfficient schemes are required for addressing flow problems in geologically complex media
Schemes well known in the literature for meeting many of the previous criteria are the following: Mixed Finite Element methods, Control-Volume Finite Element methods, Mimetic Finite Difference methods, Cell-Centered Finite Volume methods, Multipoint Flux Approximation and Discrete Duality Finite Volume methods (DDFV methods for short)
The first formulation is based on interface flux computations for primary and dual meshes, accounting with the interface flux continuity and the second formulation of DDFV is based on pressure gradient reconstructions over a diamond grid

Summary

Introduction and the Model Problem

Efficient schemes are required for addressing flow problems in geologically complex media. The first formulation is based on interface flux computations for primary and dual meshes, accounting with the interface flux continuity (see, e.g., [20, 21]) and the second formulation of DDFV is based on pressure gradient reconstructions over a diamond grid (see [22,23,24]) Note that this second formulation attracted the attention of some mathematicians as Andreianov, Boyer, and Hubert who have greatly contributed to its mathematical development. For presenting our analysis of DDFV method, let us consider the 2D diffusion problem consisting of finding a function φ which satisfies the following partial differential equation associated with nonhomogeneous Neumann boundary conditions:. (iii) Let us emphasize that the novelty of this work is that Neumann boundary conditions are imposed on the whole boundary, which causes additional difficulties in the analysis

A Finite Volume Formulation of the Model Problem

Stability and Error Estimates

Numerical Test