Full Discretization of Cauchy's Problem by Lavrentiev-Finite Element Method

Faker Ben Belgacem (1) , Vivette Girault (2) , Faten Jelassi (1)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We conduct a detailed study of the fully discrete finite element approximation of the data completion problem. This is the continuation of [Numer. Math., 139 (2016), pp. 1--25], where the variational problem, resulting from the Kohn--Vogelius duplication framed into the Steklov--Poincaré condensation approach, was semidiscretized. Under the condition that the problem has a solution, we derive a bound of the error with respect to the mesh-size and the Lavrentiev regularization parameter. Sharp local finite element estimates, such as those derived by Nitsche and Schatz [Math. Comp., 28 (1974), pp. 937--958], are the central technical tools of the analysis.

Mots clés: Cauchy problem; ill-posed problem; Lavrentiev regularization; finite elements; full discretization

Domaines: Mathématiques [math]

Faker Ben Belgacem, Vivette Girault, Faten Jelassi. Full Discretization of Cauchy's Problem by Lavrentiev-Finite Element Method. SIAM Journal on Numerical Analysis, 2022, 60 (2), pp.558--584. ⟨10.1137/21M1401310⟩. ⟨hal-03932759⟩