Analysis using inf-sup stability

In this chapter, we revisit the well-posedness of the model parabolic problem, i.e., we give another proof of Lions’ theorem using the framework of the BNB theorem. In other words, we establish the well-posedness by proving an inf-sup condition. Then we exploit the inf-sup condition to revisit the stability and the error analysis for various approximation techniques investigated in the previous chapters: (1) the space semi-discrete problem; (2) the implicit Euler scheme; (3) the discontinuous Galerkin scheme; (4) the continuous Petrov–Galerkin scheme.