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IMA Journal of Numerical Analysis Advance Access published online on August 25, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp009
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Finite-volume schemes for noncoercive elliptic problems with Neumann boundary conditions

Claire Chainais-Hillairet{dagger}

Laboratoire de Mathématiques, UMR CNRS 6620, Université Blaise Pascal, 63177 Aubière Cedex, France

Jérôme Droniou{ddagger}

Département de Mathématiques, UMR CNRS 5149, CC 051, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier cedex 5, France

{dagger} Corresponding author. Email: claire.chainais{at}math.univ-bpclermont.fr

{ddagger} Email: droniou{at}math.univ-montp2.fr

Received on 25 July 2008. Revised on 6 January 2009.


  Abstract

We consider a convective–diffusive elliptic problem with Neumann boundary conditions. The presence of the convective term results in noncoercivity of the continuous equation and, because of the boundary conditions, the equation has a nontrivial kernel. We discretize this equation with finite-volume techniques and in a general framework that allows us to consider several treatments of the convective term, namely, via a centred scheme, an upwind scheme (widely used in fluid mechanics problems) or a Scharfetter–Gummel scheme (common to semiconductor literature). We prove that these schemes satisfy the same properties as the continuous problem (one-dimensional kernel spanned by a positive function, for instance) and that their kernel and solution converge to the kernel and solution of the partial differential equation. We also present several numerical implementations, studying the effects of the choice of one scheme or the other in the approximation of the solution or the kernel.

Key Words: convection–diffusion equations; Neumann boundary conditions; finite-volume schemes; numerical analysis


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