Inf-sup stability implies quasi-orthogonality

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the coercivity of the underlying problem. The quasi-orthogonality of Galerkin solutions is a key argument in modern proofs of optimal convergence of adaptive mesh refinement algorithms. Our generalization together with other well-understood properties of the error estimator implies linear convergence of the estimator and hence rate optimal convergence. This approach provides a simple proof of optimality for non-symmetric FEM-BEM coupling. Moreover, it allows us to prove optimal convergence of an adaptive time-stepping scheme for parabolic equations.