T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

To solve variational indefinite problems, one uses classically the Banach–Ne?as–Babu?ka theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $text{ div}sigma nabla u=f$ set in $H^1_0$ , with $sigma $ exhibiting a sign change.