Optimal computational costs of AFEM with optimal local hp-robust multigrid solver
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In this work, an adaptive finite element algorithm for symmetric second-order elliptic diffusion problems with inexact solver is developed. The discrete systems are treated by a local higher-order geometric multigrid method extending the approach of [Miraçi, Papež, Vohralík, SIAM J. Sci. Comput. (2021)]. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator is h- and p-robustly equivalent to the algebraic error. The proofs rely on suitably chosen robust stable decompositions and a strengthened Cauchy-Schwarz inequality on bisection-generated meshes. Together, this yields that the proposed adaptive algorithm has optimal computational cost. Numerical experiments confirm the theoretical findings.
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