论文标题
一般二阶椭圆PDE的一阶系统最小二乘法的强制性证明
Several Proofs of Coerciveness of First-Order System Least-Squares Methods for General Second-Order Elliptic PDEs
论文作者
论文摘要
在本文中，我们介绍了一般（可能是不确定的）二阶线性椭圆PDE的一阶系统最小二乘法的强制性证明，这是在最小的唯一性假设下的。对于一般线性的二阶椭圆PDE，由于操作员和弗雷德霍尔姆（Fredholm）的替代方案的紧凑性，独特性，存在和适应性相当。因此，仅假定一个最小的唯一性假设：均匀方程具有独特的零解决方案。不需要标准变分问题的强制性。该论文的主要贡献是我们的第一个证明，这是使用标准变异配方的Inf-Sup稳定性的简单明了的证明。一旦具有标准配方的稳定性，证明就可以可能应用于其他方程式或设置。我们还提供了另外两个证据，以了解一般二阶线性椭圆PDE的最小二乘方法。第二个证明是基于不连续的彼得 - 盖尔金方法中引入的引理，第三个证明是基于对分解问题的各种稳定性分析。作为一个应用程序，我们还讨论了最小二乘有限元方法，以提供非词性$ h^{ - 1} $右侧的问题。
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.
