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

2020/05/28 by Théophile Chaumont-Frelet, Chaumont-Frelet, Théophile, Alexandre Ern +3 · 3 citations
Engineering · #65N15 #65N30 #78M10 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2005.14537

openalex publication_date 2020/05/29 · openalex created_date 2021/05/10 · openalex updated_date 2026/07/28

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 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)-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.

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• \boldsymbolH(curl)-reconstruction of piecewise polynomial fields with application to hp-a posteriori nonconforming error analysis for Maxwell's equations
• A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds

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