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

Abstract

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 $p\ge 1$ 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.

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Cited by in corpus (3)

• Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs
• On full linear convergence and optimal complexity of adaptive FEM with inexact solver
• Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs