A-posteriori-steered h- and p-robust multigrid solvers and adaptivity
A talk in the BI.discrete Workshop series by
Ani Miraci from Vienna
| Abstract: | This is a joint work with Jan Papež (Institute of Mathematics, Czech Academy of Sciences, Czech Republic), Dirk Praetorius (TU Wien, Austria), Julian Streitberger (TU Wien, Austria), and Martin Vohralík (Inria Paris, France).
We study a symmetric second-order linear elliptic PDE discretized by piecewise polynomials of arbitrary degree $p \ge 1$. To treat the arising linear system, we propose a geometric multigrid method with zero pre- and one post-smoothing by an overlapping Schwarz (block Jacobi) method [Miraçi, Papež, and Vohralík. SIAM J. Sci. Comput. 2021]. Introducing optimal step sizes which minimize the algebraic error in the level-wise error correction step of multigrid, one obtains an explicit Pythagorean formula for the algebraic error. Importantly, this inherently induces a fully computable a posteriori estimator for the energy norm of the algebraic error.
We show the two following results and their equivalence: 1) the solver contracts the algebraic error independently of the polynomial degree $p$; 2) 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 [Schöberl, Melenk, Pechstein, and Zaglmayr. IMA J. Numer. Anal. 2008] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [Chen, Nochetto, and Xu. Numer. Math. 2012].
Moreover, recent developments in [Miraçi, Praetorius, and Streitberger. In preparation.] allow to prove that a local variant of the solver is robust also with respect to the number of mesh levels used for rate-optimal adaptive finite element method.
Finally, we present a variety of numerical tests to confirm the theoretical results and to illustrate the advantages of our approach.
Within the CRC this talk is associated to the project(s): B7 |