Inf-sup stability implies quasi-orthogonality

TL;DR

This paper establishes a new link between inf-sup stability and quasi-orthogonality, enabling simplified proofs of optimal adaptive mesh refinement algorithms for various complex PDE problems.

Contribution

It introduces a generalized quasi-orthogonality derived from inf-sup stability, removing key technical barriers in proving optimal convergence of adaptive methods.

Findings

Simplified proofs for optimality of adaptive algorithms

Application to stationary Stokes and unbounded transmission problems

New stability bounds for LU-factorization of matrices

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 tool are new stability bounds for the LU-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.