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.

翻译：在本文中,我们提出了三套第一阶系统最低方位方法对第二阶椭圆形PDE的腐蚀性证据。第一版基于对PDE的先验错误估计,其假设最弱。第二和第三份证据假定了确保标准变异配方的腐蚀性的系数的充分条件。第二份证据是简单的直接证据,第三份证据是基于不连续的Petrov-Galerkin方法中引入的脂质。通过指出不同证据的优缺点和局限性,我们希望该文件能为未来证据提供指南。作为应用,我们还讨论了与美元-美元右侧问题有关的最差方位固定要素方法。