Some p-robust a posteriori error estimates based on auxiliary spaces

2025/11/10 by Li, Yuwen
Engineering · Computer Science · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks

paper · doi:10.48550/arxiv.2511.06603

Abstract

This work develops polynomial-degree-robust (p-robust) equilibrated a posteriori error estimates for H(\rm curl), H(\rm div) and H(\rm divdiv) problems, based on H¹ auxiliary space decomposition. The proposed framework employs auxiliary space preconditioning and regular decompositions to decompose the finite element residual into H^-1 residuals that are further controlled by classical p-robust equilibrated a posteriori error analysis. As a result, we obtain novel and simple p-robust a posteriori error estimates of H(\rm curl)/H(\rm div) conforming methods and mixed methods for the biharmonic equation. In addition, we prove guaranteed a posteriori upper error bounds under convex domains or certain boundary conditions. Numerical experiments demonstrate the effectiveness and p-robustness of the proposed error estimators for the Nédélec edge element methods and the Hellan--Herrmann--Johnson methods.

Citations

• Smoother-type a posteriori error estimates for finite element methods
• A 2-complex containing Sobolev spaces of matrix fields
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• A mass conserving mixed stress formulation for the Stokes equations
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Cited by

• Smoother-type a posteriori error estimates for finite element methods

Related

• Nodal auxiliary a posteriori error estimates
• Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the biharmonic problem
• Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem
• A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds
• Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system