Singularities and Error Estimates in Non-Conforming Approximation of Two-Dimensional Di usion Problem

The solution of a two dimensional di usion problem with discontinuous coefcients is analyzed on a particular two-regions domain. The solution is shown to contain a singular part, and the particular behavior is quantitatively evaluated. It is well kown that the degree of regularity of the solution for such problem determines the accuracy of the numerical techniques used to approximate the solution. The presence of singular points will then a ect the convergence orders for numerical solution of the problem. This consequences are analyzed by deriving some error estimates for a non-conforming approximation. The real e ects of the singularity are then studied on numerical calculations for di erent non-conforming elements. We will conclude from these tests that the damages due to the singularity are not as many important as theory may provided. Rsum On etudie dans ce travail un probl eme de di usion a coe cients discontinus dans le cas particulier d'un domaine compos e de deux r egions distinctes. La solution contient alors une partie singuli ere que nous evaluons en etudiant la forme analytique du ux aux points singuliers. Sachant que le degr e de r egularit e de la fonction joue un rôle primordial sur la pr ecision des m ethodes num eriques utilis ees, nous etudions l'impact des singularit es sur les majorations d'erreur d'une approximation non conforme. Les tests num eriques compl etent cette analyse par une etude des d egradations dues aux singularit es et qui s'av erent moindres que ce que la th eorie laisse supposer. Key-words: Neutronic di usion, Singularities, error estimates, nonconforming approximation. Laboratoire de Calcul Scienti que Universit de Franche Comt e, 16 route de Gray, 25000 Besan con, France. yE.D.F./D.E.R./R.N.E./Ph.R., 1 avenue du Gnral de Gaulle, 92141 Clamart Cedex, France. email: [email protected] 1