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

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 ) \boldsymbol H(\boldsymbol {\operatorname {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 Nédélec 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 ) \boldsymbol H(\boldsymbol {\operatorname {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.