Section: New Results

Polynomial-degree-robust multilevel algebraic error estimator & solver

Participants : Ani Miraci, Jan Papež, Martin Vohralík.

In [58], we devise a novel multilevel a posteriori estimator of the algebraic error. It delivers a fully computable, guaranteed lower bound on the error between an unknown exact solution of a system of linear algebraic equations and its approximation by an algebraic solver. The bound is also proved to be efficient, i.e., it also gives an upper bound on the algebraic error. Remarkably, the quality of these bounds is independent of the approximation polynomial degree. The derived estimates give immediately rise to a multilevel iterative algebraic solver whose contraction factor is independent of the polynomial degree of the approximation. We actually prove an equivalence between efficiency of the estimator and contraction of the solver. The estimator/solver are based on a global coarsest-level solve of lowest-order ($p=1$), followed by local patchwise $p$-degree problems solved on the other levels. It corresponds to a V-cycle geometric multigrid solver with zero pre- and one post-smoothing step via block-Jacobi. A salient feature is the choice of the optimal step size for the descent direction.