In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. For general linear second-order elliptic PDEs, the uniqueness, existence, and well-posedness are equivalent due to the compactness of the operator and Fredholm alternative. Thus only a minimal uniqueness assumption is assumed: the homogeneous equation has a unique zero solution. The coerciveness of the standard variational problem is not required. The paper's main contribution is our first proof, which is a straightforward and short proof using the inf-sup stability of the standard variational formulation. The proof can potentially be applied to other equations or settings once having the standard formulation's stability. We also present two other proofs for the least-squares methods of general second-order linear elliptic PDEs. The second proof is based on a lemma introduced in the discontinuous Petrov-Galerkin method, and the third proof is based on various stability analyses of the decomposed problems. As an application, we also discuss least-squares finite element methods for problems with a nonsingular $H^{-1}$ right-hand side.
翻译:在本文中,我们提供了对一般(可能无限期)二级线性椭圆形PDE采用一阶系统最低方块方法的强制性证据。对于一般线性二阶椭圆形PDE,其独特性、存在性和充分性相当于操作员和Fredholm替代物的紧凑性。因此,仅假设一个最低的独特性假设:单一方程式有一个独特的零解决办法。标准变异问题的胁迫性不需要。文件的主要贡献是我们的第一个证据,这是使用标准变异配方的不易变式的稳定性的直截了当和简短的证据。一旦标准配方具有稳定性,该证据就可能适用于其他方块或设置。我们还为一般二阶线性线性椭圆形PDES的最小方块方法提供了另外两个证据。第二个证据是以不连续的Petrov-Galkin方法中引入的利玛为根据,而第三个证据是以各种稳定性因素分析为根据的。 一种最不稳定性的方法,也是我们不易变式的。