Several Coercivity Proofs of First-Order System Least-Squares Methods for Second-Order Elliptic PDEs

In this paper, we present three versions of proofs of the coercivity for first-order system least-squares methods for second-order elliptic PDEs. The first version is based on the a priori error estimate of the PDEs, which has the weakest assumption. For the second and third proofs, a sufficient condition on the coefficients ensuring the coercivity of the standard variational formulation is assumed. The second proof is a simple direct proof and the third proof is based on a lemma introduced in the discontinuous Petrov-Galerkin method. By pointing out the advantages and limitations of different proofs, we hope that the paper will provide a guide for future proofs. As an application, we also discuss least-squares finite element methods for problems with $H^{-1}$ righthand side.