\boldsymbolH(curl)-reconstruction of piecewise polynomial fields with application to hp-a posteriori nonconforming error analysis for Maxwell's equations

2025/08/18 by Zhaonan Dong, Dong, Zhaonan, Alexandre Ern +1
Engineering · Mathematics · #65N15 #65N30 #78M10 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2508.12904

openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We devise and analyse a novel \boldsymbolH(curl)-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. Our main focus is on homogeneous tangential boundary conditions. We prove that the difference between the reconstructed \boldsymbolH₀(curl)-field and the original, piecewise polynomial field, measured in the broken curl norm and in the \boldsymbolL²-norm, can be bounded in terms of suitable jump norms of the original field. The bounds are always h-optimal, and p-suboptimal by \frac12-order for the broken curl norm and by \frac32-order for the \boldsymbolL²-norm. An auxiliary result of independent interest is a novel broken-curl, divergence-preserving Poincaré inequality on vertex patches. Moreover, the \boldsymbolL²-norm estimate can be improved to \frac12-order suboptimality under a (reasonable) assumption on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over the vertex patches. We also discuss extensions of the \boldsymbolH₀(curl)-reconstruction operator to the prescription of mixed boundary conditions, to agglomerated polytopal meshes, and to convex domains. Finally, we showcase an important application of the \boldsymbolH(curl)-reconstruction operator to the hp-a posteriori nonconforming error analysis of Maxwell's equations. We focus on the (symmetric) interior penalty discontinuous Galerkin (dG) approximation of some simplified forms of Maxwell's equations.

Citations

• Computable Poincaré--Friedrichs constants for the L^p de~Rham complex over convex domains and domains with shellable triangulations
• A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell's equations under minimal regularity assumptions
• hp-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes
• A stable local commuting projector and optimal hp approximation estimates in \boldsymbol H(curl)
• A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes
• A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem
• Estimation of the continuity constants for Bogovski\uı and regularized Poincaré integral operators
• Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem
• Finite element quasi-interpolation and best approximation
• Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
• A posteriori error estimators for nonconforming finite element methods

Related

• p-robust equilibrated flux reconstruction in \boldsymbol H(curl) based on local minimizations. Application to a posteriori analysis of the curl-curl problem
• Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem
• Analysis of the edge finite element approximation of the Maxwell\n equations with low regularity solutions
• Regularity of vector fields with piecewise regular curl and divergence
• Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron