hp -Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. 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 which comes with the solver is hp -robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.