A-Posteriori-Steered $p$-Robust Multigrid with Optimal Step-Sizes and Adaptive Number of Smoothing Steps

Abstract

We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p ≥ 1$. Our solver employs zero pre- and one post-smoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of Schoberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of Xu et al. [Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659]. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.