Inf-sup stability implies quasi-orthogonality
Michael Feischl
Abstract
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 tools are new stability bounds for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L upper U"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">LU</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.