## December 10 – Ani Miraçi: A-posteriori-steered and adaptive p-robust multigrid solvers

Ani Miraçi: Thursday 10 December at 16:00 via this link with meeting ID: 950 9381 2553 and code: 077683 We consider systems of linear algebraic equations arising from discretizations of second-order elliptic partial differential equations using finite elements of arbitrary polynomial degree. First, we propose an a posteriori estimator of the algebraic error, the construction of which is linked to that of a multigrid solver without pre-smoothing and a single post-smoothing step by overlapping Schwarz methods (block-Jacobi). We prove the following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree (p-robustness); the estimator represents a two-sided p-robust bound of the algebraic error. Next, we introduce optimal step-sizes which minimize the error at each level. This makes it possible to have an explicit Pythagorean formula for the algebraic error; this in turn serves as the foundation for a simple and efficient adaptive strategy allowing to choose the number of post-smoothing steps per level. We also introduce an adaptive local smoothing strategy thanks to our efficient and localized estimator of the algebraic error by levels/patches of elements. Patches contributing more than a user-prescribed percentage to the overall error are marked via a bulk-chasing criterion. Each iteration is composed of a non-adaptive V-cycle and an adaptive V-cycle which uses local smoothing only in the marked patches. These V-cycles contract the algebraic error in a p-robust fashion. Finally, we also extend some the above results to a mixed finite elements setting in two space dimensions. A variety of numerical tests is presented to confirm our theoretical results and to illustrate the advantages of our approaches.