In this talk, we consider a second-order linear elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree p≥1. To treat the arising linear system, we propose a geometric multigrid method. We present in detail the construction of this algebraic solver, which uses zero pre- and one post-smoothing step by an overlapping Schwarz (block Jacobi) method. Importantly, the construction of this solver has a built-in a posteriori estimator for the algebraic error; for this reason the solver is referred to as a-posteriori-steered.
The first main result is the p-robust (i.e., independent of the polynomial degree p) algebraic error contraction of the solver at each iteration. The second main result is the p-robust efficiency of the a posteriori estimator (i.e, being a two-sided bound) of the algebraic error.
Moreover, the a posteriori error control allows to localize in which levels of the multigrid (and possibly patches of elements) the algebraic error is most important. This idea leads to two adaptive extensions of the algebraic solver.
Finally, we present a variety of numerical tests to confirm the p-robust theoretical results and to illustrate the advantages of our adaptive approaches.