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 uniform inf-sup stability 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 still implies linear convergence of the estimator and hence rate optimal convergence. This approach drastically simplifies a recent proof of optimality of the adaptive algorithm for the stationary Stokes problem and generalizes it to d\geq 2 and provides a 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.
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.
D
indefinite,
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uniform inf-sup stability
coercivity
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well understood
well-understood
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still
R
drastically simplifies a recent proof of optimality of the adaptive algorithm for the stationary Stokes problem and generalizes it to d\geq 2 and provides a
provides a simple