We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tool are new stability bounds for the $LU$-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

翻译：我们通过提议对准正反方程进行一般化的准正反方程方法,从而证明对非对称、无限期和时间依赖问题的适应性网格改进算法具有新的最佳效果。 这完全消除了适应性网格改进算法最佳趋同现代证据中的核心技术困难,并导致对固定式斯托克斯问题的泰勒-休德分解、无约束式传输问题的有限元素/约束性分解、以及对抛物线方程的适应性时间分步制方法的简单最佳效果证明。 主要技术工具是使基数的基数以美元计数与最近确定的准正反正和矩阵因子化之间的联系具有新的稳定性界限。