A Finite Element Method for the Boundary Data Recovery in an Oxygen-Balance Dispersion Model

Faker Ben Belgacem (1) , Naima Debit (2, 3) , Henda El Fekih (4) , Souad Khiari (1, 4)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 ICJ - Institut Camille Jordan
3 MMCS - Modélisation mathématique, calcul scientifique
4 LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis]

The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional transport model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the outfall point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequences of these non-standard boundary conditions is that dispersive transport equations on both oxygen tracers are strongly coupled and the resulting system becomes ill-posed. The main purpose is a finite element space-discretization of the variational problem put under a non-symmetric mixed form. Combining analytical calculations, numerical computations and theoretical justifications, we try to elucidate the characteristics related to the ill-posedness of this data completion dynamical problem and understand its mathematical structure.

Mots clés: Mixed finite elements; Differential algebraic equations.; Dispersive transport; Data completion; Ill-posedness

Mots clés: Differential algebraic equations

Domaines: Analyse numérique [math.NA]

Faker Ben Belgacem, Naima Debit, Henda El Fekih, Souad Khiari. A Finite Element Method for the Boundary Data Recovery in an Oxygen-Balance Dispersion Model. Journal of Inverse and Ill-posed Problems, 2016, 24 (5), ⟨10.1515/jiip-2014-0053⟩. ⟨hal-01023384⟩