A-POSTERIORI-STEERED p-ROBUST MULTIGRID WITH OPTIMAL STEP-SIZES AND ADAPTIVE NUMBER OF SMOOTHING STEPS

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 postsmoothing 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 levelwise and patchwise error reductions. We show the following two 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 [J. Schoberl et al., IMA T. Numer. Anal., 28 (2008), pp. 1-24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 599-659]. We consider quasi-uniform or graded bisection simplicial meshes and prove at most linear dependence on the number of mesh levels for minimal HI-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 postsmoothing 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.