Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case
Eliane Bécache Laurent Bourgeois Lucas Franceschini Jérémi Dardé
In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.
keywords: heat/wave equation with lateral Cauchy data quasi-reversibility method let-set method inverse obstacle problem Backward heat equation mixed formulation. finite element method

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