Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem

TL;DR

This paper develops stable polynomial extensions and p-robust a posteriori error estimators for the curl-curl problem, enabling reliable and efficient error control in finite element discretizations without global H(curl) conformity.

Contribution

It introduces a novel stable patchwise polynomial extension technique and applies it to create reliable, efficient, and polynomial-degree-robust a posteriori error estimators for Nédélec finite element methods.

Findings

Estimators are reliable and locally efficient.

Error estimators are polynomial-degree-robust.

Numerical experiments confirm theoretical results.

Abstract

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl-curl problem discretized with N\'ed\'elec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H(curl)-conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.