A Multilevel Algebraic Error Estimator and the Corresponding Iterative Solver with $p$-Robust Behavior

Abstract

In this work, we consider conforming finite element discretizations of arbitrary polynomial degree p ≥ 1 of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant independent of the degree p. We next design a multilevel iterative algebraic solver from our estimator and show that this solver contracts the algebraic error on each iteration by a factor bounded independently of p. Actually, we show that these two results are equivalent. The p-robustness results rely on the work of Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one given mesh. We combine this with the design of an algebraic residual lifting constructed over a hierarchy of nested unstructured, possibly highly graded, simplicial meshes. The lifting includes a global coarse-level solve with the lowest polynomial degree one together with local contributions from the subsequent mesh levels. These contributions, of the highest polynomial degree p on the finest mesh, are given as solutions of mutually independent local Dirichlet problems posed over overlapping patches of elements around vertices. The construction of this lifting can be seen as one geometric V-cycle multigrid step with zero pre-and one postsmoothing by (damped) additive Schwarz (block Jacobi). One particular feature of our approach is the optimal choice of the step-size generated from the algebraic residual lifting. Numerical tests are presented to illustrate the theoretical findings.

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

“…Also, the weighted restricted additive Schwarz construction of Definition 5.1 actually behaves better in both the numerical experiments in Sec. 9 as well as in [62,Sec. 6] than that with damping of Remark 7.5, for which the p-robust result is available.…”

Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steeredp-Robust Multigrid Solvers

“…We present here our main result for the solver introduced in Section 3. Similarly to [21,22], we show for each substep that the error contraction of the solver is equivalent to the efficiency of the associated a posteriori error estimator.…”

“…There is a strong link between the solver defined in Section 3 and the a posteriori estimators defined in Section 4. Similarly to [21,22], we have the following theorem (recall also Lemma 4.2).…”

“…Larger subdomains can also be considered, cf. [21]. Finally, denote by I pj j the P pj Lagrange interpolation operator on the mesh level j, i.e.…”

An a posteriori-based adaptive preconditioner for controlling a local algebraic error norm

“…Subsequent works used the theory of a posteriori error estimates in order to derive rigorous upper bounds of the global error, which includes the algebraic error [3,2,16,12,10,20]. More recent works focused on deriving appropriate guaranteed upper bound directly on the algebraic error using equilibrated flux reconstructions [20,17,19]. In the context of using a posteriori error estimates, an adaptive preconditioner, which is used in combination with a specific initial guess and based on the estimated local distribution of the algebraic error, is derived in [1].…”